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Mathematical Methods

ContentsArticles

Hypergeometric function 1Generalized hypergeometric function 11Sturm–Liouville theory 21Hermite polynomials 27Jacobi polynomials 40Legendre polynomials 43Chebyshev polynomials 49Gegenbauer polynomials 62Laguerre polynomials 64Eigenfunction 73

ReferencesArticle Sources and Contributors 76Image Sources, Licenses and Contributors 77

Article LicensesLicense 78

Hypergeometric function 1

Hypergeometric functionIn mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented bythe hypergeometric series, that includes many other special functions as specific or limiting cases. [It] is a solutionof a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regularsingular points can be transformed into this equation.For systematic lists of some of the many thousands of published identities involving the hypergeometric function,see the reference works by Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger et al. (1953), Abramowitz &Stegun (1965), and Daalhuis (2010).

HistoryThe term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by CarlFriedrich Gauss (1813). Studies in the nineteenth century included those of Ernst Kummer (1836), and thefundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of thedifferential equation it satisfies. Riemann showed that the second-order differential equation for 2F1(z), examined inthe complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).

The hypergeometric seriesThe hypergeometric function is defined for |z| < 1 by the power series

It is undefined (or infinite) if c equals a non-positive integer. Here (q)n is the (rising) Pochhammer symbol, which isdefined by:

The series terminates if either a or b is a nonpositive integer. For complex arguments z with |z| ≥ 1 it can beanalytically continued along any path in the complex plane that avoids the branch points 0 and 1.As c goes to a non-positive integer −m, 2F1(z) goes to infinity, but if we divide by the gamma function Γ(c), we havea limit:

2F1(z) is the most usual type of generalized hypergeometric series pFq, and is often designated simply F(z).

Hypergeometric function 2

Special casesMany of the common mathematical functions can be expressed in terms of the hypergeometric function, or aslimiting cases of it. Some typical examples are

The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometricfunction

so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits ofhypergeometric functions. These include most of the commonly used functions of mathematical physics.Legendre functions are solutions of a second order differential equation with 3 regular singular points so can beexpressed in terms of the hypergeometric function in many ways, for example

Several orthogonal polynomials, including Jacobi polynomials P(α,β)n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written interms of hypergeometric functions using

Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials,Meixner–Pollaczek polynomials.Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functionswhose arguments a, b, c are 1, 1/2, 1/3, ... or 0. For example, if

then

is an elliptic modular function of τ.Incomplete beta functions Bx(p,q) are related by

The complete elliptic integrals K and E are given by

Hypergeometric function 3

The hypergeometric differential equationThe hypergeometric function is a solution of Euler's hypergeometric differential equation

which has three regular singular points: 0,1 and ∞. The generalization of this equation to three arbitrary regularsingular points is given by Riemann's differential equation. Any second order differential equation with three regularsingular points can be converted to the hypergeometric differential equation by a change of variables.

Solutions at the singular pointsSolutions to the hypergeometric differential equation are built out of the hypergeometric series 2F1(a,b;c;z). Theequation has two linearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually twospecial solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicialequation and x is a local variable vanishing at the regular singular point. This gives 3 × 2 = 6 special solutions, asfollows.Around the point z = 0, two independent solutions are, if c is not a non-positive integer,

and, on condition that c is not an integer,

If c is a non-positive integer 1−m, then the first of these solutions doesn't exist and must be replaced byThe second solution doesn't exist when c is an integer greater than 1, and is

equal to the first solution, or its replacement, when c is any other integer. So when c is an integer, a morecomplicated expression must be used for a second solution, equal to the first solution multiplied by ln(z), plusanother series in powers of z, involving the digamma function. See Abramowitz & Stegun (1965) for details.Around z = 1, if c − a − b is not an integer, one has two independent solutions

and

Around z = ∞, if a − b is not an integer, one has two independent solutions

and

Again, when the conditions of non-integrality are not met, there exist other solutions that are more complicated.Any 3 of the above 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving (63) = 20 linear relations between them called connection formulas.

Kummer's 24 solutionsA second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter group Dn of order n!2n−1. For the hypergeometric equation n=3, so the group is of order 24 and is isomorphic to the symmetric group on 4 points, and was first described by Kummer. The isomorphism with the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an

Hypergeometric function 4

even number of singular points). Kummer's group of 24 transformations is generated by the three transformationstaking a solution F(a,b;c;z) to one of

which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4points 1, 2, 3, 4. (The first and third of these are actually equal to F(a,b;c;z) whereas the second is an independentsolution to the differential equation.)Applying Kummer's 24=6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions abovecorresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 timesbecause of the identities

Q-formThe hypergeometric differential equation may be brought into the Q-form

by making the substitution w = uv and eliminating the first-derivative term. One finds that

and v is given by the solution to

which is

The Q-form is significant in its relation to the Schwarzian derivative.[1]

Schwarz triangle mapsThe Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.

where k is one of the points 0, 1, ∞. The notation

is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps.Note that each triangle map is regular at z ∈ {0, 1, ∞} respectively, with

and

Hypergeometric function 5

In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps are conformal maps of the upper half-planeH to triangles on the Riemann sphere, bounded by circular arcs. This mapping is a special case of aSchwarz–Christoffel mapping. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of thetriangle are πλ, πμ and πν respectively.Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, and thes-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r).

Monodromy groupThe monodromy of a hypergeometric equation describes how fundamental solutions change when analyticallycontinued around paths in the z plane that return to the same point. That is, when the path winds around a singularityof 2F1, the value of the solutions at the endpoint will differ from the starting point.Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thusthe monodromy is a mapping (group homomorphism):

where π1 is the fundamental group. In other words the monodromy is a two dimensional linear representation of thefundamental group. The monodromy group of the equation is the image of this map, i.e. the group generated by themonodromy matrices.

Integral formulas

Euler typeIf B is the beta function then

provided |z| < 1 or |z| = 1 and both sides converge, and can be proved by expanding (1 − zx)−a using the binomialtheorem and then integrating term by term. This was given by Euler in 1748 and implies Euler's and Pfaff'shypergeometric transformations.Other representations, corresponding to other branches, are given by taking the same integrand, but taking the pathof integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspondto the monodromy action.

Barnes integralBarnes used the theory of residues to evaluate the Barnes integral

as

where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a − 1, ..., −b, −b − 1, ... .

Hypergeometric function 6

John transformThe Gauss hypergeometric function can be written as a John transform (Gelfand, Gindikin & Graev 2003, 2.1.2).

Gauss' contiguous relationsThe six functions

are called contiguous to 2F1(a,b;c;z). Gauss showed that 2F1(a,b;c;z) can be written as a linear combination of anytwo of its contiguous functions, with rational coefficients in terms of a,b,c, and z. This gives (62)=15 relations, given by identifying any two lines on the right hand side of

In the notation above, and so on.Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form

where m, n, and l are integers.

Gauss' continued fractionGauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as acontinued fraction, for example:

Hypergeometric function 7

Transformation formulasTransformation formulas relate two hypergeometric functions at different values of the argument z.

Fractional linear transformationsEuler's transformation is

It follows by combining the two Pfaff transformations

which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations,see papers by Rathie & Paris and Rakha & Rathie.

Quadratic transformationsIf two of the numbers 1 − c, c − 1, a − b, b − a, a + b − c, c − a − b are equal or one of them is 1/2 then there is aquadratic transformation of the hypergeometric function, connecting it to a different value of z related by aquadratic equation. The first examples were given by Kummer (1836), and a complete list was given by Goursat(1881). A typical example is

Higher order transformationsIf 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is a cubic transformation of thehypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples weregiven by Goursat (1881). A typical example is

There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c arecertain rational numbers.

Values at special points zSee (Slater 1966, Appendix III) for a list of summation formulas at special points, most of which also appear in(Bailey 1935). (Gessel & Stanton 1982) gives further evaluations at more points. (Koepf 1995) shows how most ofthese identities can be verified by computer algorithms.

Special values at z = 1Gauss's theorem, named for Carl Friedrich Gauss, is the identity

which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity, first found byZhu Shijie (= Chu Shi-Chieh), as a special case.Dougall's formula generalizes this to the bilateral hypergeometric series at z = 1.

Hypergeometric function 8

Kummer's theorem (z = −1)There are many cases where hypergeometric functions can be evaluated at z = −1 by using a quadratic transformationto change z = −1 to z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer'stheorem, named for Ernst Kummer:

which follows from Kummer's quadratic transformations

and Gauss's theorem by putting z = −1 in the first identity. For generalization of Kummer's summation, see a paperby Lavoie, et al.

Values at z = 1/2Gauss's second summation theorem is

Bailey's theorem is

For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see a paper by Lavoie,et al.

Other pointsThere are many other formulas giving the hypergeometric function as an algebraic number at special rational valuesof the parameters, some of which are listed in (Gessel & Stanton 1982) and (Koepf 1995). Some typical examplesare given by

which can be restated as

whenever −π < x < π and T is the (generalized) Chebyshev polynomial.

References[1] Hille, Einar (1976), Ordinary differential equations in the complex domain, Dover, pp. 374–401, ISBN 0-486-69620-0, Chapter 10, "The

Schwarzian".

• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 15" (http:/ / www. math. sfu. ca/ ~cbm/ aands/page_555. htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NewYork: Dover, p. 555, ISBN 978-0486612720, MR  0167642 (http:/ / www. ams. org/mathscinet-getitem?mr=0167642).

• Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications 71. Cambridge University Press. ISBN 978-0-521-62321-6. MR  1688958 (http:/ / www. ams.

Hypergeometric function 9

org/ mathscinet-getitem?mr=1688958).• Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge.• Beukers, Frits (2002), Gauss' hypergeometric function (http:/ / www. math. uu. nl/ people/ beukers/

MRIcourse93. ps). (lecture notes reviewing basics, as well as triangle maps and monodromy)• Daalhuis, Adri B. Olde (2010), "Hypergeometric function" (http:/ / dlmf. nist. gov/ 15), in Olver, Frank W. J.;

Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions,Cambridge University Press, ISBN 978-0521192255, MR 2723248 (http:/ / www. ams. org/mathscinet-getitem?mr=2723248)

• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendentalfunctions (http:/ / apps. nrbook. com/ bateman/ Vol1. pdf). Vol. I. New York–Toronto–London: McGraw–HillBook Company, Inc. ISBN 978-0-89874-206-0. MR  0058756 (http:/ / www. ams. org/mathscinet-getitem?mr=0058756).

• Gasper, George & Rahman, Mizan (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia ofMathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.

• Gauss, Carl Friedrich (1813). "Disquisitiones generales circa seriem infinitam  " (http:/ / books. google. com/ books?id=uDMAAAAAQAAJ).

Commentationes societatis regiae scientarum Gottingensis recentiores (in Latin) (Göttingen) 2. (a reprint of thispaper can be found in Carl Friedrich Gauss, Werke (http:/ / books. google. com/ books?id=uDMAAAAAQAAJ),p. 125)

• Gelfand, I. M.; Gindikin, S.G. & Graev, M.I. (2003) [2000]. Selected topics in integral geometry (http:/ / books.google. com/ books?isbn=0821829327). Translations of Mathematical Monographs 220. Providence, R.I.:American Mathematical Society. ISBN 978-0-8218-2932-5. MR  2000133 (http:/ / www. ams. org/mathscinet-getitem?mr=2000133).

• Gessel, Ira & Stanton, Dennis (1982). "Strange evaluations of hypergeometric series". SIAM Journal onMathematical Analysis 13 (2): 295–308. doi: 10.1137/0513021 (http:/ / dx. doi. org/ 10. 1137/ 0513021). ISSN 0036-1410 (http:/ / www. worldcat. org/ issn/ 0036-1410). MR  647127 (http:/ / www. ams. org/mathscinet-getitem?mr=647127).

• Goursat, Édouard (1881). "Sur l'équation différentielle linéaire, qui admet pour intégrale la sériehypergéométrique" (http:/ / www. numdam. org/ item?id=ASENS_1881_2_10__S3_0). Annales Scientifiques del'École Normale Supérieure (in French) 10: 3–142. Retrieved 2008-10-16.

• Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces.San Diego: Academic Press. ISBN 0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups)

• Klein, Felix (1981). Vorlesungen über die hypergeometrische Funktion (http:/ / resolver. sub. uni-goettingen. de/purl?PPN375394591). Grundlehren der Mathematischen Wissenschaften (in German) 39. Berlin, New York:Springer-Verlag. ISBN 978-3-540-10455-1. MR  668700 (http:/ / www. ams. org/mathscinet-getitem?mr=668700).

• Koepf, Wolfram (1995). "Algorithms for m-fold hypergeometric summation". Journal of Symbolic Computation20 (4): 399–417. doi: 10.1006/jsco.1995.1056 (http:/ / dx. doi. org/ 10. 1006/ jsco. 1995. 1056). ISSN  0747-7171(http:/ / www. worldcat. org/ issn/ 0747-7171). MR  1384455 (http:/ / www. ams. org/mathscinet-getitem?mr=1384455).

• Kummer, Ernst Eduard (1836). "Über die hypergeometrische Reihe " (http:/ / resolver. sub. uni-goettingen.

de/ purl?GDZPPN00214056X). Journal für die reine und angewandte Mathematik (in German) 15: 39–83,127–172. ISSN  0075-4102 (http:/ / www. worldcat. org/ issn/ 0075-4102).

• Lavoie,J.L., Grondin, F. and Rathie, A.K., Generalizations of Whipple's theorem on the sum of a 3F2, J. Comput.Appl. Math., 72, 293-300,(1996).

Hypergeometric function 10

• Press, W.H.; Teukolsky, S.A.; Vetterling, W.T. & Flannery, B.P. (2007). "Section 6.13. HypergeometricFunctions" (http:/ / apps. nrbook. com/ empanel/ index. html#pg=318). Numerical Recipes: The Art of ScientificComputing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

• Riemann, Bernhard (1857). "Beiträge zur Theorie der durch die Gauss'sche Reihe F(α, β, γ, x) darstellbarenFunctionen" (http:/ / gdz. sub. uni-goettingen. de/ dms/ load/ img/ ?PPN=GDZPPN002018691). Abhandlungender Mathematicshen Classe der K oniglichen Gesellschaft der Wissenschaften zu G ottingen (Göttingen: Verlagder Dieterichschen Buchhandlung) 7: 3–22. (a reprint of this paper can be found in All publications of Riemann(http:/ / www. emis. de/ classics/ Riemann/ PFunct. pdf) PDF)

• Slater, Lucy Joan (1960). Confluent hypergeometric functions. Cambridge, UK: Cambridge University Press. MR 0107026 (http:/ / www. ams. org/ mathscinet-getitem?mr=0107026).

• Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press.ISBN 0-521-06483-X. MR  0201688 (http:/ / www. ams. org/ mathscinet-getitem?mr=0201688). (there is a 2008paperback with ISBN 978-0-521-09061-2)

• Wall, H.S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc.• Whittaker, E.T. & Watson, G.N. (1927). A Course of Modern Analysis. Cambridge, UK: Cambridge University

Press.• Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration

Spaces. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. ISBN 3-528-06925-2. MR  1453580 (http:/ / www.ams. org/ mathscinet-getitem?mr=1453580).

• Rathie, Arjun K. & Paris, R.B.: An extension of the Euler's-type transformation for the 3F2 series: Far EastJ.Math.Sci., 27(1), 43-48 (2007).

• Rakha, M.A. & Rathie, Arjun K. : Extensions of Euler's type- II transformation and Saalschutz's theorem: Bull.Korean Math. Soc.,48(1), 151-156 (2011).

External links• Hazewinkel, Michiel, ed. (2001), "Hypergeometric function" (http:/ / www. encyclopediaofmath. org/ index.

php?title=p/ h048450), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4• John Pearson, Computation of Hypergeometric Functions (http:/ / people. maths. ox. ac. uk/ porterm/ research/

pearson_final. pdf) (University of Oxford, MSc Thesis)• Marko Petkovsek, Herbert Wilf and Doron Zeilberger, The book "A = B" (http:/ / www. cis. upenn. edu/ ~wilf/

AeqB. html) (freely downloadable)• Weisstein, Eric W., " Hypergeometric Function (http:/ / mathworld. wolfram. com/ HypergeometricFunction.

html)", MathWorld.

Generalized hypergeometric function 11

Generalized hypergeometric functionIn mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficientsindexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function,which may then be defined over a wider domain of the argument by analytic continuation. The generalizedhypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refersto the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometricfunction and the confluent hypergeometric function as special cases, which in turn have many particular specialfunctions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.

NotationA hypergeometric series is formally defined as a power series

in which the ratio of successive coefficients is a rational function of n. That is,

where A(n) and B(n) are polynomials in n.For example, in the case of the series for the exponential function,

,

βn = n!−1 and βn+1/βn = 1/(n+1). So this satisfies the definition with A(n) = 1 and B(n) = n + 1.It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linearfactors of the form (aj + n) and (bk + n) respectively, where the aj and bk are complex numbers.For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B canbe multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.The ratio between consecutive coefficients now has the form

,

where c and d are the leading coefficients of A and B. The series then has the form

,

or, by scaling z by the appropriate factor and rearranging,

.

This has the form of an exponential generating function. The standard notation for this series is

or

Using the rising factorial or Pochhammer symbol:

this can be written

Generalized hypergeometric function 12

(Note that this use of the Pochhammer symbol is not standard, however it is the standard usage in this context.)

Special casesMany of the special functions in mathematics are special cases of the confluent hypergeometric function or thehypergeometric function; see the corresponding articles for examples.Some of the functions related to more complicated hypergeometric functions include:• Dilogarithm:

• Hahn polynomials:

• Wilson polynomials:

TerminologyWhen all the terms of the series are defined and it has a non-zero radius of convergence, then the series defines ananalytic function. Such a function, and its analytic continuations, is called the hypergeometric function.The case when the radius of convergence is 0 yields many interesting series in mathematics, for example theincomplete gamma function has the asymptotic expansion

which could be written za−1e−z 2F0(1−a,1;;−z−1). However, the use of the term hypergeometric series is usuallyrestricted to the case where the series defines an actual analytic function.The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite itsname, is a rather more complicated and recondite series. The "basic" series is the q-analog of the ordinaryhypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including theones coming from zonal spherical functions on Riemannian symmetric spaces.The series without the factor of n! in the denominator (summed over all integers n, including negative) is called thebilateral hypergeometric series.

Convergence conditionsThere are certain values of the aj and bk for which the numerator or the denominator of the coefficients is 0.• If any aj is a non-positive integer (0, −1, −2, etc.) then the series only has a finite number of terms and is, in fact, a

polynomial of degree −aj.• If any bk is a non-positive integer (excepting the previous case with −bk < aj) then the denominators become 0 and

the series is undefined.Excluding these cases, the ratio test can be applied to determine the radius of convergence.• If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value

of z. An example is the power series for the exponential function.

Generalized hypergeometric function 13

• If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z| < 1 anddiverges for |z| > 1. Whether it converges for |z| = 1 is more difficult to determine. Analytic continuation can beemployed for larger values of z.

• If p > q + 1 then the ratio of coefficients grows without bound. This implies that, besides z = 0, the seriesdiverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for adifferential equation that the sum satisfies.

The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the seriesconverges absolutely at z = 1 if

.

Further, if p=q+1, and z is real, then the following convergence result holds (Quigley et al 2013):

.

Basic propertiesIt is immediate from the definition that the order of the parameters aj, or the order of the parameters bk can bechanged without changing the value of the function. Also, if any of the parameters aj is equal to any of theparameters bk, then the matching parameters can be "cancelled out", with certain exceptions when the parameters arenon-positive integers. For example,

.

Euler's integral transformThe following basic identity is very useful as it relates the higher-order hypergeometric functions in terms ofintegrals over the lower order ones

DifferentiationThe generalized hypergeometric function satisfies

Combining these gives a differential equation satisfied by w = pFq:

.

Generalized hypergeometric function 14

Contiguous function and related identitiesTake the following operator:

From the differentiation formulas given above, the linear space spanned by

contains each of

Since the space has dimension 2, any three of these p+q+2 functions are linearly dependent. These dependencies canbe written out to generate a large number of identities involving .For example, in the simplest non-trivial case,

,

,

,So

.

This, and other important examples,

,

,

,

,

,

can be used to generate continued fraction expressions known as Gauss's continued fraction.

Similarly, by applying the differentiation formulas twice, there are such functions contained in

which has dimension three so any four are linearly dependent. This generates more identities and the process can becontinued. The identities thus generated can be combined with each other to produce new ones in a different way.A function obtained by adding ±1 to exactly one of the parameters aj, bk in

is called contiguous to

Generalized hypergeometric function 15

Using the technique outlined above, an identity relating and its two contiguous functions can be given,six identities relating and any two of its four contiguous functions, and fifteen identities relating

and any two of its six contiguous functions have been found. (The first one was derived in theprevious paragraph. The last fifteen were given by Gauss in his 1812 paper.)

IdentitiesA number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries.

Saalschütz's theoremSaalschütz's theorem[1] (Saalschütz 1890) is

For extension of this theorem, see a research paper by Rakha & Rathie.

Dixon's identityDixon's identity,[2] first proved by Dixon (1902), gives the sum of a well-poised 3F2 at 1:

For generalization of Dixon's identity, see a paper by Lavoie, et al.

Dougall's formulaDougall's formula (Dougall 1907) gives the sum of a terminating well-poised [3] series:

provided that m is a non-negative integer (so that the series terminates) and

Many of the other formulas for special values of hypergeometric functions can be derived from this as special orlimiting cases.

Generalization of Kummer's transformations and identities for 2F2Identity 1.

where

;

Identity 2.

which links Bessel functions to 2F2; this reduces to Kummer's second formula for b = 2a:

Generalized hypergeometric function 16

Identity 3.

.

Identity 4.

which is a finite sum if b-d is a non-negative integer.

Kummer's relationKummer's relation is

Clausen's formulaClausen's formula

was used by de Branges to prove the Bieberbach conjecture.

Special cases

The series 0F0

As noted earlier, . The differential equation for this function is , which has solutions

where k is a constant.

The series 1F0Also as noted earlier,

The differential equation for this function is

or

which has solutions

where k is a constant.

is the geometric series with ratio z and coefficient 1.

Generalized hypergeometric function 17

The series 0F1The functions of the form are called confluent hypergeometric limit functions and are closely relatedto Bessel functions. The relationship is:

The differential equation for this function is

or

When a is not a positive integer, the substitution

gives a linearly independent solution

so the general solution is

where k, l are constants. (If a is a positive integer, the independent solution is given by the appropriate Besselfunction of the second kind.)

The series 1F1The functions of the form are called confluent hypergeometric functions of the first kind, alsowritten . The incomplete gamma function is a special case.The differential equation for this function is

or

When b is not a positive integer, the substitution

gives a linearly independent solution

so the general solution is

where k, l are constants.

When a is a non-positive integer, −n, is a polynomial. Up to constant factors, these are the Laguerrepolynomials. This implies Hermite polynomials can be expressed in terms of 1F1 as well.

Generalized hypergeometric function 18

The series 2F0This occurs in connection with the exponential integral function Ei(z).

The series 2F1Historically, the most important are the functions of the form . These are sometimes called Gauss's

hypergeometric functions, classical standard hypergeometric or often simply hypergeometric functions. The termGeneralized hypergeometric function is used for the functions pFq if there is risk of confusion. This function wasfirst studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence.The differential equation for this function is

or

It is known as the hypergeometric differential equation. When c is not a positive integer, the substitution

gives a linearly independent solution

so the general solution for |z| < 1 is

where k, l are constants. Different solutions can be derived for other values of z. In fact there are 24 solutions, knownas the Kummer solutions, derivable using various identities, valid in different regions of the complex plane.When a is a non-positive integer, −n,

is a polynomial. Up to constant factors and scaling, these are the Jacobi polynomials. Several other classes oforthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressedusing 2F1 as well. This includes Legendre polynomials and Chebyshev polynomials.A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.:

The series 3F1This occurs in the theory of Bessel functions. It provides a way to compute Bessel functions of large arguments.

GeneralizationsThe generalized hypergeometric function is linked to the Meijer G-function and the MacRobert E-function.Hypergeometric series were generalised to several variables, for example by Paul Emile Appell; but a comparablegeneral theory took long to emerge. Many identities were found, some quite remarkable. A generalization, theq-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenthcentury. Here, the ratios considered of successive terms, instead of a rational function of n, are a rational function ofqn. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an ellipticfunction (a doubly periodic meromorphic function) of n.

Generalized hypergeometric function 19

During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections toother fields. There are a number of new definitions of general hypergeometric functions, by Aomoto, Israel Gelfandand others; and applications for example to the combinatorics of arranging a number of hyperplanes in complexN-space (see arrangement of hyperplanes).Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces andsemi-simple Lie groups. Their importance and role can be understood through the following example: thehypergeometric series 2F1 has the Legendre polynomials as a special case, and when considered in the form ofspherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or,equivalently, the rotations given by the Lie group SO(3). In tensor product decompositions of concreterepresentations of this group Clebsch-Gordan coefficients are met, which can be written as 3F2 hypergeometricseries.Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers,not just the positive ones.Fox–Wright functions are a generalization of generalized hypergeometric functions where the Pochhammer symbolsin the series expression are generalised to gamma functions of linear expressions in the index n.

Citations[1][1] See or for a proof.[2][2] See for a detailed proof. An alternative proof is in[3] http:/ / mathworld. wolfram. com/ Well-Poised. html

References• Askey, R. A.; Daalhuis, Adri B. Olde (2010), "Generalized hypergeometric function" (http:/ / dlmf. nist. gov/ 16),

in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook ofMathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 (http:/ / www. ams.org/ mathscinet-getitem?mr=2723248)

• Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics andits Applications 71. Cambridge University Press. ISBN 978-0-521-62321-6; 978-0-521-78988-2 Check |isbn=value (help). MR  1688958 (http:/ / www. ams. org/ mathscinet-getitem?mr=1688958).

• Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and MathematicalPhysics 32. London: Cambridge University Press. Zbl  0011.02303 (http:/ / www. zentralblatt-math. org/ zmath/en/ search/ ?format=complete& q=an:0011. 02303).

• Dixon, A.C. (1902). "Summation of a certain series". Proc. London Math. Soc. 35 (1): 284–291. doi:10.1112/plms/s1-35.1.284 (http:/ / dx. doi. org/ 10. 1112/ plms/ s1-35. 1. 284). JFM  34.0490.02 (http:/ / www.zentralblatt-math. org/ zmath/ en/ search/ ?format=complete& q=an:34. 0490. 02).

• Dougall, J. (1907). "On Vandermonde's theorem and some more general expansions". Proc. Edinburgh Math.Soc. 25: 114–132. doi: 10.1017/S0013091500033642 (http:/ / dx. doi. org/ 10. 1017/ S0013091500033642).

• Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and ItsApplications 96 (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 0-521-83357-4. MR  2128719(http:/ / www. ams. org/ mathscinet-getitem?mr=2128719). Zbl  1129.33005 (http:/ / www. zentralblatt-math. org/zmath/ en/ search/ ?format=complete& q=an:1129. 33005). (the first edition has ISBN 0-521-35049-2)

• Gauss, Carl Friedrich (1813). "Disquisitiones generales circa seriam infinitam  " (http:/ / books. google. com/ books?id=uDMAAAAAQAAJ).

Commentationes societatis regiae scientarum Gottingensis recentiores (in Latin) (Göttingen) 2. (a reprint of thispaper can be found in Carl Friedrich Gauss, Werke (http:/ / books. google. com/ books?id=uDMAAAAAQAAJ),p. 125)

Generalized hypergeometric function 20

• Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces.San Diego: Academic Press. ISBN 0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups)

• Lavoie, J.L.; Grondin, F.; Rathie, A.K.; Arora, K. (1994). "Generalizations of Dixon's theorem on the sum of a3F2". Math. Comp. 62: 267–276.

• Miller, A. R.; Paris, R. B. (2011). "Euler-type transformations for the generalized hypergeometric function

r+2Fr+1". Zeit. Angew. Math. Physik: 31–45. doi: 10.1007/s00033-010-0085-0 (http:/ / dx. doi. org/ 10. 1007/s00033-010-0085-0).

• Quigley, J.; Wilson, K.J.; Walls, L.; Bedford, T. (2013). "A Bayes linear Bayes Method for Estimation ofCorrelated Event Rates". Risk Analysis. doi: 10.1111/risa.12035 (http:/ / dx. doi. org/ 10. 1111/ risa. 12035).

• Rathie, Arjun K.; Pogány, Tibor K. (2008). "New summation formula for 3F2(1/2) and a Kummer-type IItransformation of 2F2(x)" (http:/ / hrcak. srce. hr/ file/ 37118). Mathematical Communications 13: 63–66. MR 2422088 (http:/ / www. ams. org/ mathscinet-getitem?mr=2422088). Zbl  1146.33002 (http:/ / www.zentralblatt-math. org/ zmath/ en/ search/ ?format=complete& q=an:1146. 33002).

• Rakha, M.A.; Rathie, Arjun K. (2011). "Extensions of Euler's type- II transformation and Saalschutz's theorem".Bull. Korean Math. Soc. 48 (1): 151–156.

• Saalschütz, L. (1890). "Eine Summationsformel". Zeitschrift für Mathematik und Physik (in German) 35:186–188. JFM  22.0262.03 (http:/ / www. zentralblatt-math. org/ zmath/ en/ search/ ?format=complete& q=an:22.0262. 03).

• Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press.ISBN 0-521-06483-X. MR  0201688 (http:/ / www. ams. org/ mathscinet-getitem?mr=0201688). Zbl  0135.28101(http:/ / www. zentralblatt-math. org/ zmath/ en/ search/ ?format=complete& q=an:0135. 28101). (there is a 2008paperback with ISBN 978-0-521-09061-2)

• Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of ConfigurationSpaces. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. ISBN 3-528-06925-2. MR  1453580 (http:/ / www.ams. org/ mathscinet-getitem?mr=1453580).

External links• The book "A = B" (http:/ / www. cis. upenn. edu/ ~wilf/ AeqB. html), this book is freely downloadable from the

internet.•• MathWorld

• Weisstein, Eric W., " Generalized Hypergeometric Function (http:/ / mathworld. wolfram. com/GeneralizedHypergeometricFunction. html)", MathWorld.

• Weisstein, Eric W., " Hypergeometric Function (http:/ / mathworld. wolfram. com/ HypergeometricFunction.html)", MathWorld.

• Weisstein, Eric W., " Confluent Hypergeometric Function of the First Kind (http:/ / mathworld. wolfram. com/ConfluentHypergeometricFunctionoftheFirstKind. html)", MathWorld.

• Weisstein, Eric W., " Confluent Hypergeometric Limit Function (http:/ / mathworld. wolfram. com/ConfluentHypergeometricLimitFunction. html)", MathWorld.

SturmLiouville theory 21

Sturm–Liouville theoryIn mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles FrançoisSturm (1803–1855) and Joseph Liouville (1809–1882), is a real second-order linear differential equation of the form

(1)

where y is a function of the free variable x. Here the functions p(x) > 0, q(x), and w(x) > 0 are specified at the outset.In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuousderivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on(a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required tosatisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the"weight" or "density" function.The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of(1) satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S–L) problem.Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by (1) and theprescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of thisproblem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitiandifferential operator in some function space defined by boundary conditions. The resulting theory of the existenceand asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and theircompleteness in a suitable function space became known as Sturm–Liouville theory. This theory is important inapplied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partialdifferential equations that are separable.A Sturm–Liouville (S–L) problem is said to be regular if p(x), w(x) > 0, and p(x), p'(x), q(x), and w(x) are continuousfunctions over the finite interval [a, b], and have separated boundary conditions of the form

(2)

(3)

Under the assumption that the S–L problem is regular, the main tenet of Sturm–Liouville theory states that:• The eigenvalues λ1, λ2, λ3, ... of the regular Sturm–Liouville problem (1)-(2)-(3) are real and can be ordered such

that

• Corresponding to each eigenvalue λn is a unique (up to a normalization constant) eigenfunction yn(x) which hasexactly n − 1 zeros in (a, b). The eigenfunction yn(x) is called the n-th fundamental solution satisfying the regularSturm–Liouville problem (1)-(2)-(3).

• The normalized eigenfunctions form an orthonormal basis

in the Hilbert space L2([a, b], w(x)dx). Here δmn is a Kronecker delta.Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understoodin a weak sense.

SturmLiouville theory 22

Sturm–Liouville formThe differential equation (1) is said to be in Sturm–Liouville form or self-adjoint form. All second-order linearordinary differential equations can be recast in the form on the left-hand side of (1) by multiplying both sides of theequation by an appropriate integrating factor (although the same is not true of second-order partial differentialequations, or if y is a vector.)

Examples• The Bessel equation:

can be written in Sturm–Liouville form as

• The Legendre equation:

can easily be put into Sturm–Liouville form, since D(1 − x2) = −2x, so, the Legendre equation is equivalent to

•• An example using an integrating factor:

Divide throughout by x3:

Multiplying throughout by an integrating factor of

gives

which can be easily put into Sturm–Liouville form since

so the differential equation is equivalent to

•• The integrating factor for a general second order differential equation:

multiplying through by the integrating factor

and then collecting gives the Sturm–Liouville form:

or, explicitly,

SturmLiouville theory 23

Sturm–Liouville equations as self-adjoint differential operatorsThe map

can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator inthe context of functional analysis. In fact, equation (1) can be written as

This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues λ1, λ2, λ3, ... and thecorresponding eigenvectors u1, u2, u3, ... of the L operator. The proper setting for this problem is the Hilbert spaceL2([a, b], w(x) dx) with scalar product

In this space L is defined on sufficiently smooth functions which satisfy the above boundary conditions. Moreover, Lgives rise to a self-adjoint operator. This can be seen formally by using integration by parts twice, where theboundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of aSturm–Liouville operator are real and that eigenfunctions of L corresponding to different eigenvalues are orthogonal.However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident.To overcome this problem one looks at the resolvent

where z is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts tosolving the inhomogeneous equation, which can be done using the variation of parameters formula. This shows thatthe resolvent is an integral operator with a continuous symmetric kernel (the Green's function of the problem). As aconsequence of the Arzelà–Ascoli theorem this integral operator is compact and existence of a sequence ofeigenvalues αn which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectraltheorem for compact operators. Finally, note that is equivalent to .If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. Inthis case the spectrum does no longer consist of eigenvalues alone and can contain a continuous component. There isstill an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important inquantum mechanics, since the one-dimensional time-independent Schrödinger equation is a special case of a S–Lequation.

ExampleWe wish to find a function u(x) which solves the following Sturm–Liouville problem:

(4)

where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example

Observe that if k is any integer, then the function

is a solution with eigenvalue λ = k2. We know that the solutions of a S–L problem form an orthogonal basis, and weknow from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases arealways maximal (by definition) we conclude that the S–L problem in this case has no other eigenvectors.

SturmLiouville theory 24

Given the preceding, let us now solve the inhomogeneous problem

with the same boundary conditions. In this case, we must write f(x) = x in a Fourier series. The reader may check,either by integrating ∫exp(ikx)x dx or by consulting a table of Fourier transforms, that we thus obtain

This particular Fourier series is troublesome because of its poor convergence properties. It is not clear a prioriwhether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are"square-summable", the Fourier series converges in L2 which is all we need for this particular theory to function. Wemention for the interested reader that in this case we may rely on a result which says that Fourier's series convergesat every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump atπ) converges to the average of the left and right limits (see convergence of Fourier series).Therefore, by using formula (4), we obtain that the solution is

In this case, we could have found the answer using antidifferentiation. This technique yields u = (x3 − π2x)/6, whoseFourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most caseswhen the differential equation is in many variables.

Application to normal modesCertain partial differential equations can be solved with the help of S–L theory. Suppose we are interested in themodes of vibration of a thin membrane, held in a rectangular frame, 0 ≤ x ≤ L1, 0 ≤ y ≤ L2. The equation of motionfor the vertical membrane's displacement, W(x, y, t) is given by the wave equation:

The method of separation of variables suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t).For such a function W the partial differential equation becomes X"/X + Y"/Y = (1/c2)T"/T. Since the three terms ofthis equation are functions of x,y,t separately, they must be constants. For example, the first term gives X" = λX for aconstant . The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0, L1 or y = 0, L2 anddefine the simplest possible S–L eigenvalue problems as in the example, yielding the "normal mode solutions" for Wwith harmonic time dependence,

where m and n are non-zero integers, Amn are arbitrary constants, and

The functions Wmn form a basis for the Hilbert space of (generalized) solutions of the wave equation; that is, anarbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies

. This representation may require a convergent infinite sum.

SturmLiouville theory 25

Representation of solutions and numerical calculationThe Sturm–Liouville differential equation (1) with boundary conditions may be solved in practice by a variety ofnumerical methods. In difficult cases, one may need to carry out the intermediate calculations to several hundreddecimal places of accuracy in order to obtain the eigenvalues correctly to a few decimal places.1. Shooting methods.[1][2] These methods proceed by guessing a value of λ, solving an initial value problem definedby the boundary conditions at one endpoint, say, a, of the interval [a, b], comparing the value this solution takes atthe other endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary tocorrect the original value. This strategy is not applicable for locating complex eigenvalues.2. Finite difference method.3. The Spectral Parameter Power Series (SPPS) method[3] makes use of a generalization of the following fact aboutsecond order ordinary differential equations: if y is a solution which does not vanish at any point of [a,b], then thefunction

is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations ofthese two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ0

* (often λ0* = 0; it does not

need to be an eigenvalue) and any solution y0 of (1) with λ = λ0* which does not vanish on [a, b]. (Discussion below

of ways to find appropriate y0 and λ0*.) Two sequences of functions X(n)(t), X~(n)(t) on [a, b], referred to as iterated

integrals, are defined recursively as follows. First when n = 0, they are taken to be identically equal to 1 on [a, b]. Toobtain the next functions they are multiplied alternately by 1/(py0

2) and wy02 and integrated, specifically

for n odd, for n even, (5)

for n odd, for n even, (6)

when n > 0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ:

and

Then for any λ (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1).(The functions p(x) and q(x) take part in this construction through their influence on the choice of y0.)Next one chooses coefficients c0, c1 so that the combination y = c0u0 + c1u1 satisfies the first boundary condition (2).This is simple to do since X(n)(a) = 0 and X~(n)(a) = 0, for n>0. The values of X(n)(b) and X~(n)(b) provide the valuesof u0(b) and u1(b) and the derivatives u0'(b) and u1'(b), so the second boundary condition (3) becomes an equation ina power series in λ. For numerical work one may truncate this series to a finite number of terms, producing acalculable polynomial in λ whose roots are approximations of the sought-after eigenvalues.When λ= λ0, this reduces to the original construction described above for a solution linearly independent to a givenone. The representations (5),(6) also have theoretical applications in Sturm–Liouville theory.

Construction of a nonvanishing solutionThe SPPS method can, itself, be used to find a starting solution y0. Consider the equation (p y' )' = μqy; i.e., q, w, and λ are replaced in (1) by 0, −q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ0 = 0. While there is no guarantee that u0 or u1 will not vanish, the complex function y0 = u0 + iu1 will never vanish because two linearly independent solutions of a regular S–L equation cannot vanish simultaneously as a consequence of the Sturm separation theorem. This trick gives a solution y0 of (1) for the

SturmLiouville theory 26

value λ0 = 0. In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary partswhich must be discarded.

Application to PDEsFor a linear second order in one spatial dimension and first order in time of the form:

Let us apply separation of variables, which in doing we must impose that:

Then our above PDE may be written as:

Where and

Since, by definition, and are independent of time and and are independent of position ,then both sides of the above equation must be equal to a constant:

The first of these equations must be solved as a Sturm–Liouville problem. Since there is no general analytic (exact)solution to Sturm–Liouville problems, we can assume we already have the solution to this problem, that is, we havethe eigenfunctions and eigenvalues . The second of these equations can be analytically solved once theeigenvalues are known.

Where:

SturmLiouville theory 27

References[1] J. D. Pryce, Numerical Solution of Sturm–Liouville Problems, Clarendon Press, Oxford, 1993.[2] V. Ledoux, M. Van Daele, G. Vanden Berghe, "Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics,"

Comput. Phys. Comm. 180, 2009, 532–554.[3] V. V. Kravchenko, R. M. Porter, "Spectral parameter power series for Sturm–Liouville problems," Mathematical Methods in the Applied

Sciences (MMAS) 33, 2010, 459–468

Further reading• Hazewinkel, Michiel, ed. (2001), "Sturm-Liouville theory" (http:/ / www. encyclopediaofmath. org/ index.

php?title=p/ s130620), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4• Hartman, Peter (2002). Ordinary Differential Equations (2 ed.). Philadelphia: SIAM. ISBN 978-0-89871-510-1.• Polyanin, A. D. and Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2

ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN 1-58488-297-2.• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems (http:/ / www. mat. univie. ac. at/

~gerald/ ftp/ book-ode/ ). Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. (Chapter 5)• Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger

Operators (http:/ / www. mat. univie. ac. at/ ~gerald/ ftp/ book-schroe/ ). Providence: American MathematicalSociety. ISBN 978-0-8218-4660-5. (see Chapter 9 for singular S–L operators and connections with quantummechanics)

• Zettl, Anton (2005). Sturm–Liouville Theory. Providence: American Mathematical Society. ISBN 0-8218-3905-5.

Hermite polynomialsIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability,such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus;in numerical analysis as Gaussian quadrature; in finite element methods as shape functions for beams; and inphysics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systemstheory in connection with nonlinear operations on Gaussian noise. They are named after Charles Hermite (1864)[1]

although they were studied earlier by Laplace (1810) and Chebyshev (1859).[2]

DefinitionThere are two different ways of standardizing the Hermite polynomials:

(the "probabilists' Hermite polynomials"); and

(the "physicists' Hermite polynomials").These two definitions are not exactly identical; each one is a rescaling of the other,

These are Hermite polynomial sequences of different variances; see the material on variances below.The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun. The polynomials Hen are sometimes denoted by Hn,

Hermite polynomials 28

especially in probability theory, because

is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

The first six (probabilists') Hermite polynomials Hen(x).

The first eleven probabilists' Hermitepolynomials are:

The first six (physicists') Hermite polynomials Hn(x).

and the first eleven physicists' Hermitepolynomials are:

Hermite polynomials 29

PropertiesHn is a polynomial of degree n. The probabilists' version He has leading coefficient 1, while the physicists' version Hhas leading coefficient 2n.

OrthogonalityHn(x) and Hen(x) are nth-degree polynomials for n = 0, 1, 2, 3, .... These polynomials are orthogonal with respect tothe weight function (measure)

   (He)or

   (H)i.e., we have

when m ≠ n. Furthermore,

   (probabilist)

or

   (physicist).

The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

CompletenessThe Hermite polynomials (probabilist or physicist) form an orthogonal basis of the Hilbert space of functionssatisfying

in which the inner product is given by the integral including the Gaussian weight function w(x) defined in thepreceding section,

An orthogonal basis for L2(R, w(x) dx) is a complete orthogonal system. For an orthogonal system, completeness isequivalent to the fact that the 0 function is the only function ƒ ∈ L2(R, w(x) dx) orthogonal to all functions in thesystem. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicistcase) that if ƒ satisfies

for every n ≥ 0, then ƒ = 0. One possible way to do it is to see that the entire function

vanishes identically. The fact that F(it) = 0 for every t real means that the Fourier transform of ƒ(x) exp(−x2) is 0, hence ƒ is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see

Hermite polynomials 30

"Completeness relation" below).An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L2(R, w(x) dx) consists inintroducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis forL2(R).

Hermite's differential equationThe probabilists' Hermite polynomials are solutions of the differential equation

where λ is a constant, with the boundary conditions that u should be polynomially bounded at infinity. With theseboundary conditions, the equation has solutions only if λ is a non-negative integer, and up to an overall scaling, thesolution is uniquely given by u(x) = Heλ(x). Rewriting the differential equation as an eigenvalue problem

solutions are the eigenfunctions of the differential operator L. This eigenvalue problem is called the Hermiteequation, although the term is also used for the closely related equation

whose solutions are the physicists' Hermite polynomials.With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analyticfunctions Heλ(z) for λ a complex index. An explicit formula can be given in terms of a contour integral (Courant &Hilbert 1953).

Recursion relationThe sequence of Hermite polynomials also satisfies the recursion

(probabilist)Individual coefficients are related by the following recursion formula:

and a[0,0]=1, a[1,0]=0, a[1,1]=1.

(Assuming : )

(physicist)Individual coefficients are related by the following recursion formula:

and a[0,0]=1, a[1,0]=0, a[1,1]=2.The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity

(probabilist)(physicist)

or, equivalently, by Taylor expanding,

(probabilist)

Hermite polynomials 31

(physicist)

In consequence, for the m-th derivatives the following relations hold:

(probabilist)

(physicist)

It follows that the Hermite polynomials also satisfy the recurrence relation

(probabilist)(physicist)

These last relations, together with the initial polynomials H0(x) and H1(x), can be used in practice to compute thepolynomials quickly.Turán's inequalities are

Moreover, the following multiplication theorem holds:

Explicit expressionThe physicists' Hermite polynomials can be written explicitly as

for even values of n and

for odd values of n. These two equations may be combined into one using the floor function:

The probabilists' Hermite polynomials He have similar formulas, which may be obtained from these by replacing thepower of 2x with the corresponding power of (√2)x, and multiplying the entire sum by 2−n/2.

Hermite polynomials 32

Generating functionThe Hermite polynomials are given by the exponential generating function

(probabilist)

(physicist).

This equality is valid for all x, t complex, and can be obtained by writing the Taylor expansion at x of the entirefunction z → exp(−z2) (in physicist's case). One can also derive the (physicist's) generating function by usingCauchy's Integral Formula to write the Hermite polynomials as

Using this in the sum , one can evaluate the remaining integral using the calculus of residues and

arrive at the desired generating function.

Expected valuesIf X is a random variable with a normal distribution with standard deviation 1 and expected value μ then

(probabilist)The moments of the standard normal may be read off directly from the relation for even indices

where is the double factorial. Note that the above expression is a special case of the representation ofthe probabilists' Hermite polynomials as moments

Asymptotic expansionAsymptotically, as tends to infinity, the expansion

(physicist[3])

holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changingamplitude

Which, using Stirling's approximation, can be further simplified, in the limit, to

This expansion is needed to resolve the wave-function of a quantum harmonic oscillator such that it agrees with theclassical approximation in the limit of the correspondence principle.A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of thesubstitution , for , with which one has the uniform approximation

Hermite polynomials 33

Similar approximations hold for the monotonic and transition regions. Specifically, if forthen

while for with complex and bounded then

where Ai(·) is the Airy function of the first kind.

Special Values

The Hermite polynomials evaluated at zero argument are called Hermite numbers.

In terms of the probabilist's polynomials this translates to

Relations to other functions

Laguerre polynomialsThe Hermite polynomials can be expressed as a special case of the Laguerre polynomials.

(physicist)

(physicist)

Relation to confluent hypergeometric functionsThe Hermite polynomials can be expressed as a special case of the parabolic cylinder functions.

(physicist)

where is Whittaker's confluent hypergeometric function. Similarly,

(physicist)

(physicist)

where is Kummer's confluent hypergeometric function.

Hermite polynomials 34

Differential operator representationThe probabilists' Hermite polynomials satisfy the identity

where D represents differentiation with respect to x, and the exponential is interpreted by expanding it as a powerseries. There are no delicate questions of convergence of this series when it operates on polynomials, since all butfinitely many terms vanish.Since the power series coefficients of the exponential are well known, and higher order derivatives of the monomialxn can be written down explicitly, this differential operator representation gives rise to a concrete formula for thecoefficients of Hn that can be used to quickly compute these polynomials.Since the formal expression for the Weierstrass transform W is eD2, we see that the Weierstrass transform of(√2)nHen(x/√2) is xn. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into acorresponding Maclaurin series.The existence of some formal power series g(D) with nonzero constant coefficient, such that Hen(x) = g(D)xn, isanother equivalent to the statement that these polynomials form an Appell sequence−−cf. W. Since they are anAppell sequence, they are a fortiori a Sheffer sequence.

Contour integral representationThe Hermite polynomials have a representation in terms of a contour integral, as

(probabilist)

(physicist)

with the contour encircling the origin.

GeneralizationsThe (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probabilitydistribution, whose density function is

which has expected value 0 and variance 1. One may speak of Hermite polynomials

of variance α, where α is any positive number. These are orthogonal with respect to the normal probabilitydistribution whose density function is

They are given by

In particular, the physicists' Hermite polynomials are

If

Hermite polynomials 35

then the polynomial sequence whose nth term is

is the umbral composition of the two polynomial sequences, and it can be shown to satisfy the identities

and

The last identity is expressed by saying that this parameterized family of polynomial sequences is a cross-sequence.

"Negative variance"Since polynomial sequences form a group under the operation of umbral composition, one may denote by

the sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermitepolynomials of negative variance. For α > 0, the coefficients of Hen

[−α](x) are just the absolute values of thecorresponding coefficients of Hen

[α](x).These arise as moments of normal probability distributions: The nth moment of the normal distribution withexpected value μ and variance σ2 is

where X is a random variable with the specified normal distribution. A special case of the cross-sequence identitythen says that

Applications

Hermite functionsOne can define the Hermite functions from the physicists' polynomials:

Since these functions contain the square root of the weight function, and have been scaled appropriately, they areorthonormal:

and form an orthonormal basis of L2(R). This fact is equivalent to the corresponding statement for Hermitepolynomials (see above).

The Hermite functions are closely related to the Whittaker function (Whittaker and Watson, 1962) :

and thereby to other parabolic cylinder functions. The Hermite functions satisfy the differential equation:

Hermite polynomials 36

This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so thesefunctions are the eigenfunctions.

Hermite functions 0 (black), 1 (red), 2 (blue), 3 (yellow), 4 (green), and 5 (magenta).

Hermite functions 0 (black), 2 (blue), 4 (green), and 50 (magenta).

Recursion relationFollowing recursion relations of Hermite polynomials, the Hermite functions obey

as well as

Extending the first relation to the arbitrary m-th derivatives for any positive integer m leads to

This formula can be used in connection with the recurrence relations for Hen and to calculate any derivative ofthe Hermite functions efficiently.

Hermite polynomials 37

Cramér's inequalityThe Hermite functions satisfy the following bound due to Harald Cramér

for x real, where the constant K is less than 1.086435.

Hermite functions as eigenfunctions of the Fourier transformThe Hermite functions ψn(x) are a set of eigenfunctions of the continuous Fourier transform. To see this, take thephysicist's version of the generating function and multiply by exp(−x 2/2). This gives

Choosing the unitary representation of the Fourier transform, the Fourier transform of the left hand side is given by

The Fourier transform of the right hand side is given by

Equating like powers of t in the transformed versions of the left- and right-hand sides finally yields

The Hermite functions ψn(x) are thus an orthonormal basis of L2(R) which diagonalizes the Fourier transformoperator. In this case, we chose the unitary version of the Fourier transform, so the eigenvalues are (−i) n. Theensuing resolution of the identity then serves to define powers, including fractional ones, of the Fourier transform, towit a Fractional Fourier transform generalization.

Combinatorial interpretation of coefficientsIn the Hermite polynomial Hen(x) of variance 1, the absolute value of the coefficient of xk is the number of(unordered) partitions of an n-member set into k singletons and (n − k)/2 (unordered) pairs. The sum of the absolutevalues of the coefficients gives the total number of partitions into singletons and pairs, the so-called telephonenumbers

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in OEIS).These numbers may also be expressed as a special value of the Hermite polynomials

Hermite polynomials 38

Completeness relationThe Christoffel–Darboux formula for Hermite polynomials reads

Moreover, the following identity holds in the sense of distributions

where δ is the Dirac delta function, (ψn) the Hermite functions, and δ(x − y) represents the Lebesgue measure on theline y = x in R2, normalized so that its projection on the horizontal axis is the usual Lebesgue measure. Thisdistributional identity follows by letting u → 1 in Mehler's formula, valid when −1 < u < 1:

which is often stated equivalently as

The function (x, y) → E(x, y; u) is the density for a Gaussian measure on R2 which is, when u is close to 1, veryconcentrated around the line y = x, and very spread out on that line. It follows that

when ƒ, g are continuous and compactly supported. This yields that ƒ can be expressed from the Hermite functions, assum of a series of vectors in L2(R), namely

In order to prove the equality above for E(x, y; u), the Fourier transform of Gaussian functions will be used severaltimes,

The Hermite polynomial is then represented as

With this representation for Hn(x) and Hn(y), one sees that

and this implies the desired result, using again the Fourier transform of Gaussian kernels after performing thesubstitution

Hermite polynomials 39

The proof of completeness by Mehler's formula is due to N.Wiener The Fourier integral and certain of itsapplications Cambridge Univ. Press 1933 reprinted Dover 1958

Notes[1] C. Hermite: Sur un nouveau développement en série de fonctions C. R Acad. Sci. Paris 58 1864 93-100; Oeuvres II 293-303[2] P.L.Chebyshev: Sur le développement des fonctions à une seule variable Bull. Acad. Sci. St. Petersb. I 1859 193-200; Oeuvres I 501-508[3] Abramowitz, p. 508-510 (http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_508. htm), 13.6.38 and 13.5.16

References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" (http:/ / www. math. sfu. ca/ ~cbm/ aands/

page_773. htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NewYork: Dover, p. 773, ISBN 978-0486612720, MR  0167642 (http:/ / www. ams. org/mathscinet-getitem?mr=0167642).

• Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume I, Wiley-Interscience.• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental

functions. Vol. II, McGraw-Hill ( scan (http:/ / www. nr. com/ legacybooks))• Fedoryuk, M.V. (2001), "Hermite functions" (http:/ / www. encyclopediaofmath. org/ index. php?title=H/

h046980), in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal

Polynomials" (http:/ / dlmf. nist. gov/ 18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark,Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255,MR 2723248 (http:/ / www. ams. org/ mathscinet-getitem?mr=2723248)

• Laplace, P.S. (1810), Mém. Cl. Sci. Math. Phys. Inst. France 58: 279–347• Suetin, P. K. (2001), "H/h047010" (http:/ / www. encyclopediaofmath. org/ index. php?title=H/ h047010), in

Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.• Szegő, Gábor (1939, 1955), Orthogonal Polynomials, American Mathematical Society• Wiener, Norbert (1958), The Fourier Integral and Certain of its Applications, New York: Dover Publications,

ISBN 0-486-60272-9• Whittaker, E. T.; Watson, G. N. (1962), 4th Edition, ed., A Course of Modern Analysis, London: Cambridge

University Press• Temme, Nico, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley,

New York, 1996

External links• Weisstein, Eric W., " Hermite Polynomial (http:/ / mathworld. wolfram. com/ HermitePolynomial. html)",

MathWorld.• Module for Hermite Polynomial Interpolation by John H. Mathews (http:/ / math. fullerton. edu/ mathews/ n2003/

HermitePolyMod. html)

Jacobi polynomials 40

Jacobi polynomialsIn mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classicalorthogonal polynomials. They are orthogonal with respect to the weight

on the interval [-1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshevpolynomials, are special cases of the Jacobi polynomials.[1]

The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.

Definitions

Via the hypergeometric functionThe Jacobi polynomials are defined via the hypergeometric function as follows:

where is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometricfunction is finite, therefore one obtains the following equivalent expression:

Rodrigues' formulaAn equivalent definition is given by Rodrigues' formula:

Alternate expression for real argumentFor real x the Jacobi polynomial can alternatively be written as

where s ≥ 0 and n−s ≥ 0, and for integer n

and Γ(z) is the Gamma function, using the convention that:

In the special case that the four quantities n, n+α, n+β, and n+α+β are nonnegative integers, the Jacobi polynomialcan be written as

(1)

The sum extends over all integer values of s for which the arguments of the factorials are nonnegative.

Jacobi polynomials 41

Basic properties

OrthogonalityThe Jacobi polynomials satisfy the orthogonality condition

for α, β > −1.As defined, they are not orthonormal, the normalization being

Symmetry relationThe polynomials have the symmetry relation

thus the other terminal value is

DerivativesThe kth derivative of the explicit expression leads to

Differential equationThe Jacobi polynomial Pn

(α, β) is a solution of the second order linear homogeneous differential equation

Recurrence relationThe recurrent relation for the Jacobi polynomials is:

for n = 2, 3, ....

Generating functionThe generating function of the Jacobi polynomials is given by

where

and the branch of square root is chosen so that R(z, 0) = 1.

Jacobi polynomials 42

Asymptotics of Jacobi polynomialsFor x in the interior of [-1, 1], the asymptotics of Pn

(α, β) for large n is given by the Darboux formula

where

and the "O" term is uniform on the interval [ε, π-ε] for every ε > 0.The asymptotics of the Jacobi polynomials near the points ±1 is given by the Mehler–Heine formula

where the limits are uniform for z in a bounded domain.The asymptotics outside [−1, 1] is less explicit.

Applications

Wigner d-matrixThe expression (1) allows the expression of the Wigner d-matrix dj

m’,m(φ) (for 0 ≤ φ ≤ 4π) in terms of Jacobipolynomials:

Notes[1] The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent

relation is in IV.5.

Further reading• Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and

its Applications 71, Cambridge University Press, ISBN 978-0-521-62321-6; 978-0-521-78988-2 Check |isbn=value (help), MR  1688958 (http:/ / www. ams. org/ mathscinet-getitem?mr=1688958)

• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "OrthogonalPolynomials" (http:/ / dlmf. nist. gov/ 18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark,Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255,MR 2723248 (http:/ / www. ams. org/ mathscinet-getitem?mr=2723248)

Jacobi polynomials 43

External links• Weisstein, Eric W., " Jacobi Polynomial (http:/ / mathworld. wolfram. com/ JacobiPolynomial. html)",

MathWorld.

Legendre polynomialsIn mathematics, Legendre functions are solutions to Legendre's differential equation:

They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered inphysics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partialdifferential equations) in spherical coordinates.The Legendre differential equation may be solved using the standard power series method. The equation has regularsingular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is aninteger, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates(i.e. it is a polynomial).These solutions for n = 0, 1, 2, ... (with the normalization Pn(1) = 1) form a polynomial sequence of orthogonalpolynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. Itmay be expressed using Rodrigues' formula:

That these polynomials satisfy the Legendre differential equation (1) follows by differentiating (n+1) times bothsides of the identity

and employing the general Leibniz rule for repeated differentiation. The Pn can also be defined as the coefficients ina Taylor series expansion:

In physics, this generating function is the basis for multipole expansions.

Recursive definitionExpanding the Taylor series in equation (1) for the first two terms gives

for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylorseries, equation (1) is differentiated with respect to t on both sides and rearranged to obtain

Replacing the quotient of the square root with its definition in (1), and equating the coefficients of powers of t in theresulting expansion gives Bonnet’s recursion formula

This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generatedrecursively.

Legendre polynomials 44

Explicit representations include

where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simplemonomials and involves the multiplicative formula of the binomial coefficient.The first few Legendre polynomials are:

n

0

1

2

3

4

5

6

7

8

9

10

The graphs of these polynomials (up to n = 5) are shown below:

Legendre polynomials 45

OrthogonalityAn important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner producton the interval −1 ≤ x ≤ 1:

(where δmn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise). In fact, an alternative derivation ofthe Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1, x, x2, ...} with respectto this inner product. The reason for this orthogonality property is that the Legendre differential equation can beviewed as a Sturm–Liouville problem, where the Legendre polynomials are eigenfunctions of a Hermitiandifferential operator:

where the eigenvalue λ corresponds to n(n + 1).

Applications of Legendre polynomials in physicsThe Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre[1] as the coefficients in theexpansion of the Newtonian potential

where and are the lengths of the vectors and respectively and is the angle between those two vectors.The series converges when . The expression gives the gravitational potential associated to a point mass orthe Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, forinstance, when integrating this expression over a continuous mass or charge distribution.Legendre polynomials occur in the solution of Laplace equation of the potential, , in a charge-freeregion of space, using the method of separation of variables, where the boundary conditions have axial symmetry (nodependence on an azimuthal angle). Where is the axis of symmetry and is the angle between the position of theobserver and the axis (the zenith angle), the solution for the potential will be

and are to be determined according to the boundary condition of each problem.[2]

They also appear when solving Schrödinger equation in three dimensions for a central force.Legendre polynomials in multipole expansions

Legendre polynomials 46

Figure 2

Legendre polynomials are also useful in expanding functions of theform (this is the same as before, written a little differently):

which arise naturally in multipole expansions. The left-hand side of theequation is the generating function for the Legendre polynomials.

As an example, the electric potential (in sphericalcoordinates) due to a point charge located on the z-axis at (Figure 2) varies like

If the radius r of the observation point P is greater than a, the potentialmay be expanded in the Legendre polynomials

where we have defined η = a/r < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in theLegendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipoleexpansion.

Additional properties of Legendre polynomialsLegendre polynomials are symmetric or antisymmetric, that is

[]

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials'definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by beingscaled so that

The derivative at the end point is given by

As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursionformula

and

Useful for the integration of Legendre polynomials is

From the above one can see also that

or equivalently

Legendre polynomials 47

where is the norm over the interval −1 ≤ x ≤ 1

From Bonnet’s recursion formula one obtains by induction the explicit representation

The Askey–Gasper inequality for Legendre polynomials reads

Shifted Legendre polynomialsThe shifted Legendre polynomials are defined as . Here the "shifting" function

(in fact, it is an affine transformation) is chosen such that it bijectively maps the interval [0, 1] to theinterval [−1, 1], implying that the polynomials are orthogonal on [0, 1]:

An explicit expression for the shifted Legendre polynomials is given by

The analogue of Rodrigues' formula for the shifted Legendre polynomials is

The first few shifted Legendre polynomials are:

n

0 1

1

2

3

4

Legendre polynomials 48

Legendre functionsAs well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series.These are the Legendre functions of the second kind, denoted by .

The differential equation

has the general solution

,where A and B are constants.

Legendre functions of fractional orderLegendre functions of fractional order exist and follow from insertion of fractional derivatives as defined byfractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. Theresulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regularat the endpoints. The fractional order Legendre function Pn agrees with the associated Legendre polynomial P0n.

Notes[1] M. Le Gendre, "Recherches sur l'attraction des sphéroïdes homogènes," Mémoires de Mathématiques et de Physique, présentés à l'Académie

Royale des Sciences, par divers savans, et lus dans ses Assemblées, Tome X, pp. 411-435 (Paris, 1785). [Note: Legendre submitted hisfindings to the Academy in 1782, but they were published in 1785.] Available on-line (in French) at: http:/ / edocs. ub. uni-frankfurt. de/volltexte/ 2007/ 3757/ pdf/ A009566090. pdf .

[2] Jackson, J.D. Classical Electrodynamics, 3rd edition, Wiley & Sons, 1999. page 103

References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 8" (http:/ / www. math. sfu. ca/ ~cbm/ aands/

page_332. htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NewYork: Dover, p. 332, ISBN 978-0486612720, MR  0167642 (http:/ / www. ams. org/mathscinet-getitem?mr=0167642) See also chapter 22 (http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_773. htm).

• Bayin, S.S. (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 2.• Belousov, S. L. (1962), Tables of normalized associated Legendre polynomials, Mathematical tables 18,

Pergamon Press.• Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience

Publischer, Inc.• Dunster, T. M. (2010), "Legendre and Related Functions" (http:/ / dlmf. nist. gov/ 14), in Olver, Frank W. J.;

Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions,Cambridge University Press, ISBN 978-0521192255, MR 2723248 (http:/ / www. ams. org/mathscinet-getitem?mr=2723248)

• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "OrthogonalPolynomials" (http:/ / dlmf. nist. gov/ 18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark,Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255,MR 2723248 (http:/ / www. ams. org/ mathscinet-getitem?mr=2723248)

• Refaat El Attar (2009), Legendre Polynomials and Functions, CreateSpace, ISBN 978-1-4414-9012-4

Legendre polynomials 49

External links• A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen

(http:/ / www. physics. drexel. edu/ ~tim/ open/ hydrofin)• Hazewinkel, Michiel, ed. (2001), "Legendre polynomials" (http:/ / www. encyclopediaofmath. org/ index.

php?title=p/ l058050), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4• Wolfram MathWorld entry on Legendre polynomials (http:/ / mathworld. wolfram. com/ LegendrePolynomial.

html)• Module for Legendre Polynomials by John H. Mathews (http:/ / math. fullerton. edu/ mathews/ n2003/

LegendrePolyMod. html)• Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics (http:/ / www.

du. edu/ ~jcalvert/ math/ legendre. htm)• The Legendre Polynomials by Carlyle E. Moore (http:/ / www. morehouse. edu/ facstaff/ cmoore/ Legendre

Polynomials. htm)• Legendre Polynomials from Hyperphysics (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ math/ legend. html)

Chebyshev polynomialsIn mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonalpolynomials which are related to de Moivre's formula and which can be defined recursively. One usuallydistinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomialsof the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the nameChebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials ofeither kind composes a polynomial sequence.Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the conditionthat their absolute value is bounded on the interval by 1. They are also the extremal polynomials for many otherproperties.[2]

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials ofthe first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resultinginterpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is closeto the polynomial of best approximation to a continuous function under the maximum norm. This approximationleads directly to the method of Clenshaw–Curtis quadrature.In the study of differential equations they arise as the solution to the Chebyshev differential equations

and

for the polynomials of the first and second kind, respectively. These equations are special cases of theSturm–Liouville differential equation.

Chebyshev polynomials 50

DefinitionThe Chebyshev polynomials of the first kind are defined by the recurrence relation

The conventional generating function for Tn is

The exponential generating function is

The generating function relevant for 2-dimensional potential theory and multipole expansion is

The Chebyshev polynomials of the second kind are defined by the recurrence relation

One example of a generating function for Un is

Trigonometric definitionThe Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying

or, in other words, as the unique polynomials satisfying

for n = 0, 1, 2, 3, ... which is a variant (equivalent transpose) of Schröder's equation, viz. Tn(x) is functionallyconjugate to nx, codified in the nesting property below. Further compare to the spread polynomials, in the sectionbelow.The polynomials of the second kind satisfy:

which is structurally quite similar to the Dirichlet kernel :

That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one sideof de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers ofsin(x) are even and thus replaceable through the identity cos2(x) + sin2(x) = 1.This identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one tocalculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.Evaluating the first two Chebyshev polynomials,

Chebyshev polynomials 51

and

one can straightforwardly determine that

and so forth.Two immediate corollaries are the composition identity (or nesting property specifying a semigroup)

and the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,

Pell equation definitionThe Chebyshev polynomials can also be defined as the solutions to the Pell equation

in a ring R[x].[3] Thus, they can be generated by the standard technique for Pell equations of taking powers of afundamental solution:

Relation between Chebyshev polynomials of the first and second kindsThe Chebyshev polynomials of the first and second kind are closely related by the following equations

, where n is odd.

, where n is even.

The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations

This relationship is used in the Chebyshev spectral method of solving differential equations.Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:

Chebyshev polynomials 52

These can be derived from the trigonometric formulae; for example, if , then

Note that both these equations and the trigonometric equations take a simpler form if we, like some works, followthe alternate convention of denoting our Un (the polynomial of degree n) with Un+1 instead.Turán's inequalities for the Chebyshev polynomials are

and

Explicit expressionsDifferent approaches to defining Chebyshev polynomials lead to different explicit expressions such as:

Chebyshev polynomials 53

where is a hypergeometric function.

Properties

Roots and extremaA Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in theinterval [−1,1]. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodesbecause they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that

one can easily prove that the roots of Tn are

Similarly, the roots of Un are

The extrema of Tn on the interval −1 ≤ x ≤ 1 are located at

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of theextrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the definingproperty of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at theendpoints, given by:

Chebyshev polynomials 54

Differentiation and integrationThe derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in theirtrigonometric forms, it's easy to show that:

The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form,specifically) at x = 1 and x = −1. It can be shown that:

Proof

The second derivative of the Chebyshev polynomial of the first kind is

which, if evaluated as shown above, poses a problem because it is indeterminate at x = ±1. Since the function is apolynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression aboveshould yield the desired value:

where only is considered for now. Factoring the denominator:

Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and

The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e.which will be useful later on. Since the numerator and denominator are both limiting to

zero, L'Hôpital's rule applies:

Chebyshev polynomials 55

The proof for is similar, with the fact that being important.

Indeed, the following, more general formula holds:

This latter result is of great use in the numerical solution of eigenvalue problems.Concerning integration, the first derivative of the Tn implies that

and the recurrence relation for the first kind polynomials involving derivatives establishes that

OrthogonalityBoth the Tn and the Un form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonalwith respect to the weight

on the interval [−1,1], i.e. we have:

This can be proven by letting x = cos (θ) and using the defining identity Tn(cos (θ)) = cos (nθ).Similarly, the polynomials of the second kind are orthogonal with respect to the weight

on the interval [−1,1], i.e. we have:

(Note that the measure is, to within a normalizing constant, the Wigner semicircle distribution).

Chebyshev polynomials 56

The Tn also satisfy a discrete orthogonality condition:

where the xk are the N Gauss–Lobatto zeros of TN(x)

Minimal ∞-normFor any given n ≥ 1, among the polynomials of degree n with leading coefficient 1,

is the one of which the maximal absolute value on the interval [−1, 1] is minimal.This maximal absolute value is

and |ƒ(x)| reaches this maximum exactly n + 1 times at

Proof

Let's assume that is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the

interval [−1, 1] less than .

Define

Because at extreme points of we have

From the intermediate value theorem, has at least n roots. However, this is impossible, as is apolynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.

Chebyshev polynomials 57

Other propertiesThe Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselvesare a special case of the Jacobi polynomials:

•

•

For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functionsof x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. Infact,

and

The leading coefficient of Tn is 2n − 1 if 1 ≤ n, but 1 if 0 = n.Tn are a special case of Lissajous curves with frequency ratio equal to n.Several polynomial sequences like Lucas polynomials (Ln), Dickson polynomials(Dn), Fibonacci polynomials(Fn)are related to Chebyshev polynomials Tn and Un.The Chebyshev polynomials of the first kind satisfy the relation

which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfythe similar relation

Similar to the formula

we have the analogous formula

.For ,

and

,which follows from the fact that this holds by definition for .Let

.

Then and are commuting polynomials:,

as is evident in the Abelian nesting property specified above.

Chebyshev polynomials 58

Examples

The first few Chebyshev polynomials of the first kind in the domain −1 < x < 1:The flat T0, T1, T2, T3, T4 and T5.

The first few Chebyshev polynomials of thefirst kind are A028297

The first few Chebyshev polynomials of the second kind in the domain −1 < x < 1:The flat U0, U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n +

1 and Un(−1) = (n + 1)(−1)n.

The first few Chebyshev polynomials of thesecond kind are

Chebyshev polynomials 59

As a basis set

The non-smooth function (top) y = −x3H(−x), where His the Heaviside step function, and (bottom) the 5th

partial sum of its Chebyshev expansion. The 7th sum isindistinguishable from the original function at the

resolution of the graph.

In the appropriate Sobolev space, the set of Chebyshevpolynomials form a orthonormal basis, so that a function in thesame space can, on −1 ≤ x ≤ 1 be expressed via the expansion:

Furthermore, as mentioned previously, the Chebyshev polynomialsform an orthogonal basis which (among other things) implies thatthe coefficients an can be determined easily through the applicationof an inner product. This sum is called a Chebyshev series or aChebyshev expansion.

Since a Chebyshev series is related to a Fourier cosine seriesthrough a change of variables, all of the theorems, identities, etc.that apply to Fourier series have a Chebyshev counterpart. Theseattributes include:

• The Chebyshev polynomials form a complete orthogonalsystem.

• The Chebyshev series converges to ƒ(x) if the function ispiecewise smooth and continuous. The smoothness requirementcan be relaxed in most cases — as long as there are a finitenumber of discontinuities in ƒ(x) and its derivatives.

•• At a discontinuity, the series will converge to the average of theright and left limits.

The abundance of the theorems and identities inherited fromFourier series make the Chebyshev polynomials important tools innumeric analysis; for example they are the most popular generalpurpose basis functions used in the spectral method, often in favorof trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still aproblem).

Example 1

Consider the Chebyshev expansion of . One can express

One can find the coefficients either through the application of an inner product or by the discrete orthogonalitycondition. For the inner product,

which gives

Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discreteorthogonality condition gives

Chebyshev polynomials 60

where is the Kronecker delta function and the are the N Gauss–Lobatto zeros of

This allows us to compute the coefficients very efficiently through the discrete cosine transform

Example 2To provide another example:

Partial sumsThe partial sums of

are very useful in the approximation of various functions and in the solution of differential equations (see spectralmethod). Two common methods for determining the coefficients an are through the use of the inner product as inGalerkin's method and through the use of collocation which is related to interpolation.As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on theChebyshev–Gauss–Lobatto[4] points (or Lobatto grid), which results in minimum error and avoids Runge'sphenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest orderpolynomial in the sum, plus the endpoints and is given by:

Polynomial in Chebyshev formAn arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such apolynomial p(x) is of the form

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Chebyshev polynomials 61

Spread polynomialsThe spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one toavoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.

Notes[1] Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom de parallélogrammes,"

Mémoires des Savants étrangers présentés à l’Académie de Saint-Pétersbourg, vol. 7, pages 539–586.[2] Rivlin, Theodore J. The Chebyshev polynomials. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New

York-London-Sydney,1974. Chapter 2, "Extremal Properties", pp. 56--123.[3] Jeroen Demeyer Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields (http:/ / cage. ugent. be/

~jdemeyer/ phd. pdf), Ph.D. theses (2007), p.70.[4] Chebyshev Interpolation: An Interactive Tour (http:/ / www. joma. org/ images/ upload_library/ 4/ vol6/ Sarra/ Chebyshev. html)

References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" (http:/ / www. math. sfu. ca/ ~cbm/ aands/

page_773. htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NewYork: Dover, p. 773, ISBN 978-0486612720, MR  0167642 (http:/ / www. ams. org/mathscinet-getitem?mr=0167642).

• Dette, Holger (1995), A Note on Some Peculiar Nonlinear Extremal Phenomena of the Chebyshev Polynomials(http:/ / journals. cambridge. org/ download. php?file=/ PEM/ PEM2_38_02/ S001309150001912Xa. pdf&code=c12b9a2fc1aba9e27005a5003ed21b36), Proceedings of the Edinburgh Mathematical Society 38, 343-355

• Eremenko, A.; Lempert, L. (1994), An Extremal Problem For Polynomials (http:/ / www. math. purdue. edu/~eremenko/ dvi/ lempert. pdf), Proceedings of the American Mathematical Society, Volume 122, Number 1,191-193

• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "OrthogonalPolynomials" (http:/ / dlmf. nist. gov/ 18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark,Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255,MR 2723248 (http:/ / www. ams. org/ mathscinet-getitem?mr=2723248)

• Remes, Eugene, On an Extremal Property of Chebyshev Polynomials (http:/ / www. math. technion. ac. il/ hat/fpapers/ remeztrans. pdf)

• Suetin, P.K. (2001), "C/c021940" (http:/ / www. encyclopediaofmath. org/ index. php?title=C/ c021940), inHazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

External links• Weisstein, Eric W., " Chebyshev Polynomial of the First Kind (http:/ / mathworld. wolfram. com/

ChebyshevPolynomialoftheFirstKind. html)", MathWorld.• Module for Chebyshev Polynomials by John H. Mathews (http:/ / math. fullerton. edu/ mathews/ n2003/

ChebyshevPolyMod. html)• Chebyshev Interpolation: An Interactive Tour (http:/ / www. joma. org/ images/ upload_library/ 4/ vol6/ Sarra/

Chebyshev. html), includes illustrative Java applet.• Numerical Computing with Functions: The Chebfun Project (http:/ / www. maths. ox. ac. uk/ chebfun/ )• Is there an intuitive explanation for an extremal property of Chebyshev polynomials? (http:/ / mathoverflow. net/

questions/ 25534/ is-there-an-intuitive-explanation-for-an-extremal-property-of-chebyshev-polynomia)

Gegenbauer polynomials 62

Gegenbauer polynomialsIn mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α)n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. Theygeneralize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They arenamed after Leopold Gegenbauer.

CharacterizationsA variety of characterizations of the Gegenbauer polynomials are available.• The polynomials can be defined in terms of their generating function (Stein & Weiss 1971, §IV.2):

• The polynomials satisfy the recurrence relation (Suetin 2001):

•• Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):

When α = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to theLegendre polynomials.

• They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:

(Abramowitz & Stegun p. 561 [1]). Here (2α)n is the rising factorial. Explicitly,

• They are special cases of the Jacobi polynomials (Suetin 2001):

in which represents the rising factorial of .One therefore also has the Rodrigues formula

Gegenbauer polynomials 63

Orthogonality and normalizationFor a fixed α, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz &Stegun p. 774 [2])

To wit, for n ≠ m,

They are normalized by

ApplicationsThe Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potentialtheory and harmonic analysis. The Newtonian potential in Rn has the expansion, valid with α = (n − 2)/2,

When n = 3, this gives the Legendre polynomial expansion of the gravitational potential. Similar expressions areavailable for the expansion of the Poisson kernel in a ball (Stein & Weiss 1971).

It follows that the quantities are spherical harmonics, when regarded as a function of x only.

They are, in fact, exactly the zonal spherical harmonics, up to a normalizing constant.Gegenbauer polynomials also appear in the theory of Positive-definite functions.The Askey–Gasper inequality reads

References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" [3], Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 773, ISBN 978-0486612720, MR 0167642[4].

• Bayin, S.S. (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 5.• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal

Polynomials" [5], in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NISTHandbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248 [6]

• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.:Princeton University Press, ISBN 978-0-691-08078-9.

• Suetin, P.K. (2001), "Ultraspherical polynomials" [7], in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4.

Gegenbauer polynomials 64

References[1] http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_561. htm[2] http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_774. htm[3] http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_773. htm[4] http:/ / www. ams. org/ mathscinet-getitem?mr=0167642[5] http:/ / dlmf. nist. gov/ 18[6] http:/ / www. ams. org/ mathscinet-getitem?mr=2723248[7] http:/ / www. encyclopediaofmath. org/ index. php?title=U/ u095030

Laguerre polynomialsIn mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886), are solutions ofLaguerre's equation:

which is a second-order linear differential equation. This equation has nonsingular solutions only if n is anon-negative integer.The associated Laguerre polynomials (also named Sonin polynomials after Nikolay Yakovlevich Sonin in someolder books) are solutions of

The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form

These polynomials, usually denoted L0, L1, ..., are a polynomial sequence which may be defined by the Rodriguesformula,

reducing to the closed form of a following section.They are orthogonal polynomials with respect to an inner product

The sequence of Laguerre polynomials n! Ln is a Sheffer sequence, d⁄dx Ln = (d⁄dx−1) Ln−1.The Rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementarychanges of variables.The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equationfor a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanicsin phase space. They further enter in the quantum mechanics of the 3D isotropic harmonic oscillator.Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of n! than thedefinition used here. (Likewise, some physicist may use somewhat different definitions of the so-called associatedLaguerre polynomials.)

Laguerre polynomials 65

The first few polynomialsThese are the first few Laguerre polynomials:

n

0

1

2

3

4

5

6

The first six Laguerre polynomials.

Recursive definition, closed form, and generating functionOne can also define the Laguerre polynomials recursively, defining the first two polynomials as

and then using the following recurrence relation for any k ≥ 1:

The closed form is

Laguerre polynomials 66

The exponential generating function for them likewise follows,

Generalized Laguerre polynomialsFor arbitrary real α the polynomial solutions of the differential equation [1]

are called generalized Laguerre polynomials, or associated Laguerre polynomials.The simple Laguerre polynomials are included in the associated polynomials, through α = 0,

The Rodrigues formula for them is

Explicit examples and properties of the associated Laguerre polynomials• Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as[2]

When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression[3]

in terms of Kummer's function of the second kind.• The closed form for these associated Laguerre polynomials of degree n is[4]

derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.•• The first few generalized Laguerre polynomials are:

• The coefficient of the leading term is (−1)n/n!;• The constant term, which is the value at 0, is

• Ln(α) has n real, strictly positive roots (notice that is a Sturm chain), which are all in the

interval [citation needed]

• The polynomials' asymptotic behaviour for large n, but fixed α and x > 0, is given by[5][6]

Laguerre polynomials 67

and summarizing by

where is the Bessel function.Moreover[citation needed]

,

whenever n tends to infinity.

Recurrence relationsThe addition formula for Laguerre polynomials:[7]

.

Laguerre's polynomials satisfy the recurrence relations

in particular

and

or

moreover

They can be used to derive the four 3-point-rules

Laguerre polynomials 68

combined they give this additional, useful recurrence relations

A somewhat curious identity, valid for integer i and n, is

it may be used to derive the partial fraction decomposition

Derivatives of generalized Laguerre polynomialsDifferentiating the power series representation of a generalized Laguerre polynomial k times leads to

This points to a special case (α = 0) of the formula above: for integer α = k the generalized polynomial may be

written , the shift by k sometimes causing confusion with the usual parenthesis

notation for a derivative.Moreover, this following equation holds

which generalizes with Cauchy's formula to

The derivate with respect to the second variable α has the surprising form

The generalized associated Laguerre polynomials obey the differential equation

Laguerre polynomials 69

which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

where for this equation only.

In Sturm–Liouville form the differential equation is

which shows that Lαn is an eigenvector for the eigenvalue n.

OrthogonalityThe associated Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting functionxα e −x:[8]

which follows from

If denoted the Gamma distribution then the orthogonality relation can be written as

The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)[citation needed]

recursively

Moreover,

in the associated L2[0, ∞)-space.Turán's inequalities can be derived here, which is

The following integral is needed in the quantum mechanical treatment of the hydrogen atom,

Laguerre polynomials 70

Series expansionsLet a function have the (formal) series expansion

Then

The series converges in the associated Hilbert space , iff

Further examples of expansions

Monomials are represented as

while binomials have the parametrization

This leads directly to

for the exponential function. The incomplete gamma function has the representation

Multiplication theoremsErdélyi gives the following two multiplication theorems [9]

•

•

Laguerre polynomials 71

As a contour integralGiven the generating function specified above, the polynomials may be expressed in terms of a contour integral

where the contour circles the origin once in a counterclockwise direction.

Relation to Hermite polynomialsThe generalized Laguerre polynomials are related to the Hermite polynomials:

and

where the Hn(x) are the Hermite polynomials based on the weighting function exp(−x2), the so-called "physicist'sversion."Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

Relation to hypergeometric functionsThe Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluenthypergeometric functions, as

where is the Pochhammer symbol (which in this case represents the rising factorial).

Poisson Kernel

Notes[1] A&S p. 781[2] A&S p.509[3] A&S p.510[4] A&S p. 775[5] G. Szegő, "Orthogonal polynomials", 4th edition, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975, p. 198.[6] D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", SIAM J. Numer. Anal., vol. 46 (2008), no. 6, pp. 3285-3312,

http:/ / dx. doi. org/ 10. 1137/ 07068031X[7] A&S equation (22.12.6), p. 785[8] A&S p. 774[9] C. Truesdell, " On the Addition and Multiplication Theorems for the Special Functions (http:/ / www. pnas. org/ cgi/ reprint/ 36/ 12/ 752.

pdf)", Proceedings of the National Academy of Sciences, Mathematics, (1950) pp.752-757.

Laguerre polynomials 72

References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" (http:/ / www. math. sfu. ca/ ~cbm/ aands/

page_773. htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NewYork: Dover, p. 773, ISBN 978-0486612720, MR  0167642 (http:/ / www. ams. org/mathscinet-getitem?mr=0167642).

• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "OrthogonalPolynomials" (http:/ / dlmf. nist. gov/ 18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark,Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255,MR 2723248 (http:/ / www. ams. org/ mathscinet-getitem?mr=2723248)

• B. Spain, M.G. Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970.Chapter 10 deals with Laguerre polynomials.

• Hazewinkel, Michiel, ed. (2001), "Laguerre polynomials" (http:/ / www. encyclopediaofmath. org/ index.php?title=p/ l057310), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Eric W. Weisstein, " Laguerre Polynomial (http:/ / mathworld. wolfram. com/ LaguerrePolynomial. html)", FromMathWorld—A Wolfram Web Resource.

• George Arfken and Hans Weber (2000). Mathematical Methods for Physicists. Academic Press.ISBN 0-12-059825-6.

• S. S. Bayin (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 3.

External links• Timothy Jones. "The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of the

Hydrogen Atom" (http:/ / www. physics. drexel. edu/ ~tim/ open/ hydrofin).• Weisstein, Eric W., " Laguerre polynomial (http:/ / mathworld. wolfram. com/ LaguerrePolynomial. html)",

MathWorld.

Eigenfunction 73

Eigenfunction

This solution of the vibrating drum problem is, atany point in time, an eigenfunction of the Laplace

operator on a disk.

In mathematics, an eigenfunction of a linear operator, A, defined onsome function space, is any non-zero function f in that space thatreturns from the operator exactly as is, except for a multiplicativescaling factor. More precisely, one has

for some scalar, λ, the corresponding eigenvalue. The solution of thedifferential eigenvalue problem also depends on any boundaryconditions required of . In each case there are only certaineigenvalues ( ) that admit a correspondingsolution for (with each belonging to the eigenvalue ) when combined with the boundaryconditions. Eigenfunctions are used to analyze .For example, is an eigenfunction for the differential operator

for any value of , with corresponding eigenvalue . If boundary conditions are applied to thissystem (e.g., at two physical locations in space), then only certain values of satisfy the boundaryconditions, generating corresponding discrete eigenvalues .Specifically, in the study of signals and systems, the eigenfunction of a system is the signal which when inputinto the system, produces a response with the complex constant .[1]

Examples

Derivative operatorA widely used class of linear operators acting on function spaces are the differential operators on function spaces. Asan example, on the space of infinitely differentiable real functions of a real argument , the process ofdifferentiation is a linear operator since

for any functions and in , and any real numbers and .The eigenvalue equation for a linear differential operator in is then a differential equation

The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator , aneigenfunction is a function that, when differentiated, yields a constant times the original function. That is,

for all . This equation can be solved for any value of . The solution is an exponential function

The derivative operator is defined also for complex-valued functions of a complex argument. In the complex versionof the space , the eigenvalue equation has a solution for any complex constant . The spectrum of theoperator is therefore the whole complex plane. This is an example of a continuous spectrum.

Eigenfunction 74

Applications

Vibrating strings

The shape of a standing wave in a string fixed at its boundaries is anexample of an eigenfunction of a differential operator. The admissibleeigenvalues are governed by the length of the string and determine the

frequency of oscillation.

Let denote the sideways displacement of astressed elastic chord, such as the vibrating stringsof a string instrument, as a function of the position

along the string and of time . From the lawsof mechanics, applied to infinitesimal portions ofthe string, one can deduce that the function satisfies the partial differential equation

which is called the (one-dimensional) waveequation. Here is a constant that depends on the tension and mass of the string.

This problem is amenable to the method of separation of variables. If we assume that can be written as theproduct of the form , we can form a pair of ordinary differential equations:

and

Each of these is an eigenvalue equation, for eigenvalues and , respectively. For any values of and , the equations are satisfied by the functions

and

where and are arbitrary real constants. If we impose boundary conditions (that the ends of the string are fixedwith at and , for example) we can constrain the eigenvalues. For those boundaryconditions, we find

, and so the phase angle and

Thus, the constant is constrained to take one of the values , where is any integer. Thus, theclamped string supports a family of standing waves of the form

From the point of view of our musical instrument, the frequency is the frequency of the th harmonic, which iscalled the th overtone.

Eigenfunction 75

Quantum mechanicsEigenfunctions play an important role in many branches of physics. An important example is quantum mechanics,where the Schrödinger equation

,with

has solutions of the form

where are eigenfunctions of the operator with eigenvalues . The fact that only certain eigenvalues with associated eigenfunctions satisfy Schrödinger's equation leads to a natural basis for quantum mechanics andthe periodic table of the elements, with each an allowable energy state of the system. The success of thisequation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20thcentury physics.Due to the nature of Hermitian Operators such as the Hamiltonian operator , its eigenfunctions are orthogonalfunctions. This is not necessarily the case for eigenfunctions of other operators (such as the example mentionedabove). Orthogonal functions , have the property that

where is the complex conjugate of

whenever , in which case the set is said to be orthogonal. Also, it is linearly independent.

Notes[1] Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 49

References• Methods of Mathematical Physics by R. Courant, D. Hilbert ISBN 0-471-50447-5 (Volume 1 Paperback) ISBN

0-471-50439-4 (Volume 2 Paperback) ISBN 0-471-17990-6 (Hardback)

Article Sources and Contributors 76

Article Sources and ContributorsHypergeometric function  Source: http://en.wikipedia.org/w/index.php?oldid=578455206  Contributors: 2001:638:906:2:219:D1FF:FE06:5CD6, A. Pichler, Alidev, Andrewrp,Anythingyouwant, Arjunkumarrathie, Ben Standeven, Bongwarrior, CRGreathouse, Camrn86, CitingAllArticles, ColdPhage, Connor Behan, D.Lazard, DVdm, Decaluwe.t, Delaszk, Dmcq,Dmharvey, Duoduoduo, Eric Kvaalen, Fintor, Fsedit, Giftlite, GuidoGer, Headbomb, InverseHypercube, Killing Vector, Kilom691, Kjetil1001, Linas, Lovibond, Malik Shabazz, Mathcop,MelbourneStar, Michael Hardy, MichaelPenk, Mild Bill Hiccup, MostafaSabry, Ntsimp, Oleg Alexandrov, Policron, R.Zorn, R.e.b., ServiceAT, Slawekb, Smack, Soku56, Specfunfan,Suslindisambiguator, Tavilis, Ttwo, Yecril, 85 anonymous edits

Generalized hypergeometric function  Source: http://en.wikipedia.org/w/index.php?oldid=567833658  Contributors: A. Pichler, Abelzaal, Ahoerstemeier, Algebraist, Almit39, AperiodicOrder,Arjunkrathie, Asympt, Ben Standeven, BenFrantzDale, Bo Jacoby, C h fleming, Charles Matthews, Christian.fritz, Crisófilax, Cyp, DVdm, Dagoberto.salazar, DavidCBryant, Delius, Dogaroon,Don Warren, Dysprosia, Giftlite, Hgrosser, Humanist bd, Humus sapiens, InverseHypercube, Jarry1250, Jitse Niesen, JohnBlackburne, JohnCD, JonMcLoone, Jonon, Ligulem, LilHelpa, Linas,Marcika, McKay, Michael Hardy, Msh210, Nabla, NymphadoraTonks, Omnipaedista, PAR, Quadrescence, Quadricode, R'n'B, R.Zorn, R.e.b., RDBury, Rathemis, Reedy, Rjwilmsi, Robinh,Rory-Mulvaney, Sameenahmedkhan, Schmock, Sketchjoy, Sodin, Specfunfan, Stevenj, ThePI, Thomas Bliem, Timeroot, Tinabeana, TomyDuby, Waltpohl, Wile E. Heresiarch, WriterHound,Xdb11112, Yecril, 91 anonymous edits

Sturm–Liouville theory  Source: http://en.wikipedia.org/w/index.php?oldid=579169637  Contributors: Abdull, Adam the brute, Amingare, Andrei Polyanin, ArséniureDeGallium, Bencherlite,Boatsstock, Charles Matthews, Chokoboii, Chubby Chicken, Courcelles, Danger, Debator of mathematics, DennisMYeh, Dojarca, Dysprosia, Edgar181, Emperor Tony X. Liu III, Emperor TonyX. Liu IV, Emperor Tony X. Liu VI, Emperor Tony X. Liu VII, Esperant, Favonian, FrozenUmbrella, Fuse809, Gadykozma, Gareth Owen, Gene Ray (Cubic), Giftlite, Gulloar, Hgkamath, J04n,JabberWok, Jitse Niesen, Justin545, Killing Vector, Klemensk, Larsobrien, Loisel, MathKnight, MathMartin, Mathuvw, Meisam, Meldraft, Michael Hardy, Mike409, Mr Travelling Wave Tube,Netheril96, Neutiquam, Oleg Alexandrov, Parkyere, Paulnwatts, Pavon, Policron, Pope Nigel the porter, Pope Tony Liu, Quiet Silent Bob, RMPK, Randomguess, Richard Molnár-Szipai,Rjwilmsi, Rwalker, Shotwell, Smota123, Sodin, Stevenj, Sławomir Biały, Tommy2010, Tony X Liu, Uetz, WISo, Warbler271, Wcherowi, Wikispaghetti, Wikomidia, Wile E. Heresiarch,WriterHound, Xaosflux, Xmath, Zurishaddai, 91 anonymous edits

Hermite polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=578283139  Contributors: 11kravitzn, 81120906713, A. Pichler, Adselsum, Ahoerstemeier, Alejo2083, AtroX Worf,Avraham, AxelBoldt, Baccyak4H, Bdmy, Berian, BigJohnHenry, Cuzkatzimhut, Cyp, David Eppstein, Diegotorquemada, Dysprosia, Episanty, Eruantalon, Evilbu, Genjix, Giftlite,HowiAuckland, Initialfluctuation, JFB80, JP.Martin-Flatin, Janus Antoninus, Jheald, John C PI, JohnBlackburne, Jtyard, Jusdafax, Jérôme, Kandersonovsky, Kurt.hewett, LSzilard, LaMenta3,LaguerreLegendre, Lethe, Linas, Looxix, Lumidek, MaciejDems, MarkSweep, Mathsci, Mbset, Michael Hardy, Ohconfucius, Oleg Alexandrov, Orbatar, PAR, PeterBFZ, Petr Kopač, Piovac,Polnian, Quarague, Quietbritishjim, R.e.b., Rea5, Renevets, Rjwilmsi, Ronhjones, Sachdevasushant, Silly rabbit, Slawekb, Stevenj, StewartMH, SuperSmashley, Titus III, Tobias Bergemann,TomyDuby, Topology Expert, Twri, Uffishbongo, Verdy p, William Ackerman, Witger, Zaslav, 107 anonymous edits

Jacobi polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=565226511  Contributors: A. Pichler, Almit39, Asymptoticus, BahramH, CapitalR, Charles Matthews, Cronholm144,GeordieMcBain, Giftlite, Headbomb, HowiAuckland, Ligulem, Linas, MarkSweep, Michael Hardy, P.wormer, PAR, PigFlu Oink, R.e.b., Schmock, Sodin, Thenub314, Tkuvho, Water-vole,Wikeithpedia, William Ackerman, 21 anonymous edits

Legendre polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=575556157  Contributors: A. Pichler, Abalenkm, Adselsum, Ahmes, Alejo2083, Anamitra Palit, Anand0gc,Andres.felipe.ordonez, Appraiser, Arbitrarily0, AxelBoldt, Baccyak4H, BigJohnHenry, Bosoxrock, Boud, CYD, Catslash, Celebratedsummer, Charles Matthews, Cwkmail, David Eppstein,Digitiki, Dominus, Dysprosia, ElTchanggo, En0ct9cr, Fibonacci, FvdP, Geek1337, Giftlite, Guido Kanschat, HowiAuckland, InverseHypercube, JamesBWatson, Jarble, Jarich, Jheald, JitseNiesen, JohnOwens, Keenan Pepper, Kiefer.Wolfowitz, Kotasik, LaguerreLegendre, Lambiam, Linas, Lionelbrits, MarkSweep, Maxal, Michael Hardy, Mmeijeri, Nixdorf, ObsessiveMathsFreak,Oleg Alexandrov, PAR, PaD, Policron, Pyotrnator, R.e.b., Rea5, Rjw62, Sandycx, Shadowjams, Spvo, Stamcose, Stannered, StewartMH, Sławomir Biały, Tarquin, Tercer, Toby Bartels, Trey56,Ulner, Vanished user 9i39j3, Varuna, Wile E. Heresiarch, William Ackerman, WillowW, Xnn, Yardimsever, Yerpo, Zhuman, 103 anonymous edits

Chebyshev polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=574881046  Contributors: A. Pichler, Aadri, Arthur Rubin, Auclairde, AugPi, Aymatth2, B jonas, BRicaud,Balabiot, Ben pcc, Berland, BigJohnHenry, Bochev, Brad7777, Cenarium, Charles Matthews, ChrisHodgesUK, Clausen, Cuzkatzimhut, Cwkmail, Cyp, DARTH SIDIOUS 2, David Eppstein,Dionysostom, Dmn, Dylan Thurston, Dysprosia, E Wing, El Roih, Elwikipedista, F3et, Fuse809, GDirichlet, Gaius Cornelius, Giftlite, Happy-marmotte, Headbomb, HenningThielemann,Ichbin-dcw, Illahnou, Inductiveload, J6w5, Jaredwf, Jbergquist, Jdgilbey, Jitse Niesen, Jnestorius, Justin W Smith, Kaimbridge, KoenDelaere, Kwantus, Lababidi, Leperous, Linas, Ling.Nut,Loisel, MarkSweep, MathMartin, Maxal, Mhovdan, Michael Hardy, NOrbeck, NathanHurst, Nixdorf, Oleg Alexandrov, Oliver Jennrich, Onaraighl, PAR, Piil, Pjacklam, Plastikspork, Qiqi.wang,R.e.b., Rajb245, Raoul NK, Rdengler, Régis B., Salgueiro, Slawekb, Stevenj, StewartMH, Sun Creator, Sławomir Biały, Tardis, Ting ganZ, Tobias Bergemann, Tom harrison, Tosha, Touriste,Uskudargideriken, Uzdzislaw, Vituzzu, Wild Lion, William Ackerman, Wwoods, XJaM, 186 anonymous edits

Gegenbauer polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=560374457  Contributors: 2405:B000:600:262:0:0:36:7B, A. Pichler, Adselsum, Alberto da Calvairate, Cjthellama,David Eppstein, Gagelman, Ligulem, Linas, Mark viking, MarkSweep, Michael Hardy, PAR, R.e.b., Robinh, Slawekb, Sławomir Biały, William Ackerman, 25 anonymous edits

Laguerre polynomials  Source: http://en.wikipedia.org/w/index.php?oldid=574361235  Contributors: 2001:638:504:C00E:214:22FF:FE49:D786, 2001:6B0:1:1290:969:9A13:9F7F:2845,5colourmap, A. Pichler, Adselsum, Alejo2083, Almit39, Alwayssunny1212, AugPi, Bexing, Blotwell, BryanD, Charles Matthews, Chris Howard, Chzz, Crackling, Cuzkatzimhut,Donarreiskoffer, Dysprosia, Edsanville, Freiddie, GeordieMcBain, Giftlite, Ichbin-dcw, John C PI, Jujutacular, LaguerreLegendre, Lambiam, Ligulem, Linas, Lovibond, Magioladitis,MarkSweep, Mateyone, Michael Hardy, Netheril96, Nimdaz, Oleg Alexandrov, P.wormer, PAR, Policron, R. J. Mathar, R.e.b., RDBury, Rea5, Rememberlands, Rmjarvis, Slawekb, Spastas,Stevenj, StewartMH, That Guy, From That Show!, TheEternalVortex, TomyDuby, William Ackerman, 65 anonymous edits

Eigenfunction  Source: http://en.wikipedia.org/w/index.php?oldid=558018652  Contributors: 123Mike456Winston789, Algebraist, Andrevruas, AptitudeDesign, BenFrantzDale, CES1596,Charles Matthews, Djsven, Dotbenjamin, Eagleclaw6, EdH, Evergreen9, Everyone Dies In the End, Ghyazdian, Giftlite, Hankwang, Heber A. Tzoc, Jorge Stolfi, Jurohi, Metacomet, MichaelHardy, Netheril96, Oleg Alexandrov, Paquitotrek, Pcap, RauniLillemets, Rausch, Sackofcatfood, SeventyThree, Shalom Yechiel, Silly rabbit, Simxp, Sterrys, Sullivan.t.j, Tijfo098, WISo, Wdwd,WhiteHatLurker, Wshun, Yurigerhard, 老 陳, 22 anonymous edits

Image Sources, Licenses and Contributors 77

Image Sources, Licenses and ContributorsImage:Hermite poly.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Hermite_poly.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: AlessioDamatoImage:Hermite poly phys.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Hermite_poly_phys.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:Hermite_poly_solid.svg: *Hermite_poly.svg: Alessio Damato derivative work: Vulpecula (talk) derivative work: Vulpecula (talk)Image:Herm5.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Herm5.svg  License: Public Domain  Contributors: Maciej DemsImage:Herm50.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Herm50.svg  License: Public Domain  Contributors: Maciej DemsFile:Legendrepolynomials6.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Legendrepolynomials6.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Geek3File:Point axial multipole.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Point_axial_multipole.svg  License: GNU Free Documentation License  Contributors: Traced byUser:StanneredFile:Chebyshev Polynomials of the 1st Kind (n=0-5, x=(-1,1)).svg  Source: http://en.wikipedia.org/w/index.php?title=File:Chebyshev_Polynomials_of_the_1st_Kind_(n=0-5,_x=(-1,1)).svg License: Public Domain  Contributors: InductiveloadFile:OEISicon light.svg  Source: http://en.wikipedia.org/w/index.php?title=File:OEISicon_light.svg  License: Public Domain  Contributors: Billinghurst, Mate2code, Senator2029File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg  Source: http://en.wikipedia.org/w/index.php?title=File:Chebyshev_Polynomials_of_the_2nd_Kind_(n=0-5,_x=(-1,1)).svg License: Public Domain  Contributors: InductiveloadImage:ChebyshevExpansion.png  Source: http://en.wikipedia.org/w/index.php?title=File:ChebyshevExpansion.png  License: Public Domain  Contributors: Ben pccImage:Laguerre poly.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Laguerre_poly.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: AlessioDamatoImage:Drum vibration mode12.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Drum_vibration_mode12.gif  License: Public Domain  Contributors: Oleg AlexandrovFile:Standing wave.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Standing_wave.gif  License: Public Domain  Contributors: BrokenSegue, Cdang, Joolz, Kersti Nebelsiek,LucasVB, Mike.lifeguard, Pieter Kuiper, Ptj

License 78

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