The Associated Classical Orthogonal Polynomials

Mizan Rahman ∗

School of Mathematics & Statistics, Carleton UniversityOttawa, ON K1S 5B6

Abstract

The associated orthogonal polynomials {pn(x; c)} are defined by the 3-term re-currence relation with coefficients An, Bn, Cn for {pn(x)} with c = 0, replaced byAn+c, Bn+c and Cn+c, c being the association parameter. Starting with exampleswhere such polynomials occur in a natural way some of the well-known theories of howto determine their measures of orthogonality are discussed. The highest level of thefamily of classical orthogonal polynomials, namely, the associated Askey-Wilson poly-nomials which were studied at length by Ismail and Rahman in 1991 is reviewed withspecial reference to various connected results that exist in the literature.

AMS subject classification: Primary 33A65; secondary 42C05

Key words and phrases: Classical orthogonal polynomials; Associated orthogonal poly-nomials; Associated Legendre, Laguerre, Hermite, Jacobi, q-ultraspherical, q-Jacobi andAskey-Wilson polynomials; Continued fractions, Stieltjes transform, Perron-Stieltjes inver-sion formula; Hypergeometric and basic hypergeometric series.

∗Supported in part by NSERC grant #A6197

1

1 Introduction

For problems in heat conduction or potential theory that have axial symmetry the Laplaceequation in cylindrical coordinates is

1r

∂

∂r

(r∂V

∂r

)+∂2V

∂z2= 0.(1.1)

Discrete analogues of Laplace equation were considered by Courant, Friedricks and Lewy[11]. Boyer [7] studied the solutions of the following discretization of eqn. (1.1):

(m+

12

)k

(V (mh+ h, nk) − V (mh, nk)

h

)(1.2)

−(m− 1

2

)k

(V (mh, nk) − V (mh− h, nk)

h

)

+ mh

(V (mh, nk+ k) − 2V (mh, nk) + V (mh, nk− k)

k

)= 0,

m, n = 1, 2, . . ., (h, k) represents the mesh-size, and the “boundary” condition is

V (h, nk) − V (0, nk)h

+h

4

(V (0, nk+ k) − 2V (0, nk) + V (0, nk− k)

k2

)= 0.(1.3)

It is well-known that by separating the r and z variables in (1.1) one gets the Besselfunction for the r-equation. Analogously, Boyer [7] sought solutions of (1.2) in the form

V (mh, nk) = φmenξ ,(1.4)

where ξ is related to the separation constant λ by

λ =(h

ksinh

ξ

2

)2

.(1.5)

Assuming that φ0 = 1 this leads to the 3-term recurrence relation

(2m+ 1)φm+1 + (2m− 1)φm−1 − 4m(1 − 2λ)φm = 0(1.6)

withφ0 = 1, φ1 = 1 − λ.(1.7)

Changing the variable by λ = 1−x2 , (1.6) and (1.7) become

(n+

12

)pn+1(x) = 2nxpn(x) −

(n− 1

2

)pn−1(x), n ≥ 1,(1.8)

p0(x) = 1, p1(x) =1 + x

2, φn(λ) = pn

(1 − x

2

).(1.9)

2

Boyer calls any solution of (1.8) a discrete Bessel function. However, it is clear that (1.8)is the ν = −1

2 special case of the 3- term recurrence relation of the so-called associatedLegendre polynomials:

(n+ ν + 1)Pn+1(ν, x) = (2n+ 2ν + 1)xPn(ν, x) − (n+ ν)Pn−1(ν, x),(1.10)

see also Chihara [9].In general, the 3-term recurrence relation satisfied by an orthogonal polynomial system

{pn(x)} (OPS) is of the form

pn+1(x) = (Anx+Bn)pn(x) −CnPn−1(x), n = 0, 1, . . . ,(1.11)

with p−1(x) = 0, p0(x) = 1. If AnAn+1Cn+1 > 0, n = 0, 1, . . ., then there exists a positivemeasure dµ(x) with finite moments such that the following orthogonality relation∫ ∞

−∞pn(x)pm(x)dµ(x) = λnδm,n, m, n = 0, 1, . . . , λ = A−1

0 ,(1.12)

holds.Favard’s theorem [9] assures us that the converse is also true. Finding the measure

and the explicit solutions of (1.11) are a formidable task except in some special casesincluding the very important class of what is called the classical orthogonal polynomials.For instance, the Jacobi polynomials, P (α,β)

n (x), the ultraspherical polynomial Cλn(x), theLaguerre polynomial, Lαn(x), and the Hermite polynomials, Hn(x), satisfy, respectively, therecurrence relations

P(α,β)n+1 (x) =

{(2n+ α+ β + 1)(2n+ α + β + 2)

2(n+ 1)(n+ α+ β + 1)x(1.13)

+(α2 − β2)(2n+ α+ β + 1)

2(n+ 1)(n+ α+ β + 1)(2n+ α+ β)

}P (α,β)n (x)

− (n+ α)(n+ β)(2n+ α + β + 2)(n+ 1)(n+ α+ β + 1)(2n+ α+ β)

P(α,β)n−1 (x),

Cλn+1(x) =2(n+ λ)n+ 1

xCλn(x) − n+ 2λ− 1n + 1

Cλn−1(x),(1.14)

Lαn+1(x) =2n+ α + 1 − x

n + 1Lαn(x) − n+ α

n + 1Lαn−1(x),(1.15)

Hn+1(x) = 2xHn(x) − 2nHn−1(x),(1.16)

withP (α,β)n (x) =

(α+ 1)nn! 2F1(−n, n+ α+ β + 1; α+ 1; (1 − x)/2),(1.17)

Cλn (x) =n∑k=0

(λ)k(λ)n−kk! (n− k)!

cos(n− 2k)θ(1.18)

=(2λ)nn! 2F1(−n, n+ 2λ; λ +

12

; (1 − x)/2), cos θ = x;

3

Lαn(x) =(α+ 1)n

n! 1F1(−n; α+ 1; x),(1.19)

Hn(x) = n![n2]∑

k=0

(−1)k(2x)n−2k

k! (n− 2k)!.(1.20)

Discrete classical OPS that include the Krawtchouk, Meixner and Charlier polynomials, see[13], have also similar simple expressions in terms of hypergeometric functions. Usually themeasures of orthogonality of the classical OPS can be determined in a fairly straightforwardmanner, often by quite elementary methods. However, things can change quite dramaticallyeven on an apparently harmless and innocent-looking alteration of the 3-term recurrencerelation:

pn+1(x) = (An+cx+Bn+c)pn(x) − Cn+cpn−1(x), n = 0, 1, . . . ,(1.21)

with p−1(x) = 0, p0(x) = 1, An+cAn+c+1Cn+c+1 > 0, n = 0, 1, . . ., where c is an additionalparameter such as c = −1/2 that was used in (1.8). Polynomial solutions of (1.21) satisfyingthe given initial conditions are called the associated orthogonal polynomials (i.e. associatedwith the ones with c = 0).

Apart from their possible use in physical models, exemplified by the discretized Laplaceeqn. (1.2), they occur in a natural way, for positive integer values of c, in the theory of thesolutions of the original system satisfying (1.11).

It is well-known that closely related to (1.11) is the continued fraction

F (x) =1

∣∣∣∣∣∣A0x+B0

−C1

∣∣∣∣∣∣A1x+B1

−C2

∣∣∣∣∣∣A2x+B2

− . . . ,(1.22)

see, for example, [9] and [2]. The nth convergent of this continued fraction, say, P (1)n (x)/Pn(x),

is a rational function where the denominator polynomial Pn(x) satisfies (1.11) with the sameinitial conditions, but the numerator polynomial P (1)

n (x) satisfy the same recurrence rela-tion, but a different initial condition, namely,

P(1)0 (x) = 0, P

(1)1 (x) = 1.(1.23)

Hence P(1)n (x) is a polynomial of precise degree n − 1, and so it satisfies the associated

equation

rn+1(x) = (An+1x+Bn+1)rn(x) − Cn+1rn−1(x),(1.24)

with r−1(x) = 0, r0(x) = 1, rn−1(x) = P(1)n (x). In this manner one can generate a whole

sequence of polynomial systems P (k)n (x), the so-called numerator polynomials, [9], satisfying

the recurrence relation

rn+1(x) = (An+kx+Bn+k) rn(x) −Cn+krn−1(x),(1.25)

with r−1(x) = 0, r0(x) = 1, k = 1, 2, . . ..

4

Whether or not the continued fraction in (1.22) converges it is clear that P (1)n (x)/Pn(x)

provides an [n − 1/n] Pade approximant to F (x). However, when the continued fractiondoes converge, we write

F (x) = limn→∞

P(1)n (x)Pn(x)

.(1.26)

Wimp [50] considered the [n/n] Pade approximant, An(z)/Bn(z), for the logarithmic deriva-tive of the confluent hypergeometric function 1F1(a; b; z) and used the polynomials An(z)and Bn(z) to derive a discrete orthogonality relation for the Lommel and the correspondingassociated polynomials.

The associated orthogonal polynomials also appear in a natural way in the birth-and-death processes, studied by Karlin and McGregor [28], [29], see also Ismail, Letessier andValent [22]. Let

pmn(t) = Prob{X(t) = n|X(0) = m},(1.27)

the transition probabilities of a birth-and-death process, be given by

pmn(t) =

λmt + o(t), n = m+ 1µmt + o(t), n = m− 11 − (λm + µm)t+ o(t), n = m,

(1.28)

where λn and µn are the birth and death rates at the state n, respectively. With λn > 0,µn+1 > 0, n ≥ 0, µ0 ≥ 0, Karlin and McGregor [28] proved that

pmn(t) = πn

∫ ∞

0e−xtQm(x)Qn(x)dµ(x),(1.29)

where π0 = 1, πn = (λ0 λ1 . . . λn−1)/(µ1 µ2 . . . µn), n > 0, and {Qn(x)} are polynomialsorthogonal w.r.t. dµ. If we set Fn(x) = πnQn(x), then the orthogonality relation for {Fn(x)}is ∫ ∞

0Fn(x)Fm(x)dµ(x) = πnδmn,(1.30)

where the polynomials Fn(x) satisfy the 3-term recurrence relation

Fn+1(x) =λn + µn − x

µn+1Fn(x) − λn−1

µn+1Fn−1(x),(1.31)

with

F0(x) = 1, F1(x) = (λ0 + µ0 − x)/µ1.(1.32)

For linear birth-and-death models Ismail, Letessier and Valent [22] use the two cases:(i) λn = n + α + c+ 1, µn = n + c, n ≥ 0, c > 0; (ii) λn = n+ α + c+ 1, n ≥ 0; µ0 = 0,µn = n+ c, n ≥ 1. In case (i) the recurrence relation (1.31) can be rewritten in the form

Lαn+1(x; c) =2n+ 2c+ α+ 1 − x

n + c+ 1Lαn(x; c)− n + α+ c

n + c+ 1Lαn−1(x; c),(1.33)

5

which is the relation associated to (1.15) with n replaced by n+ c in the coefficients. Askeyand Wimp [5] found the measure of orthogonality for these associated Laguerre polynomialsLαn(x; c), as well as their explicit polynomial form:

Lαn(x; c) =(α+ 1)n

n!

n∑k=0

(−n)k xk

(c+ 1)k(α+ 1)k3F2

[k − n, k + 1 − α, c

k + c+ 1, −α− n; 1

].(1.34)

By a different method the measures of orthogonality were found in [22], in addition to theexplicit form of the second system of associated Laguerre polynomials that corresponds tocase (ii):

Lαn(x; c) =(α+ 1)n

n!

n∑k=0

(−n)kxk

(c+ 1)k(α+ 1)k3F2

[k − n, k − α, ck + c+ 1, −α− n

; 1

].(1.35)

In a later paper the same authors [23] considered the polynomials associated withsymmetric birth and death processes with quadratic rates: λn = (n + a)(n + b), n ≥ 0,µn = (n + α)(n+ β), n > 0, µ0 = 0 or µ0 = αβ. They obtained the absolutely continuousmeasure for the resulting associated continuous dual Hahn orthogonal polynomials as wellas the following explicit formula:

Pn(x; a, b, α, β, η)(1.36)

=(a)nn!

n∑k=0

(−n)k(a+ γ + i√x− γ2)k(a+ γ − i

√x− γ2)k

(a)k(α+ 1)k(β + 1)k

×k∑j=0

(α)j(β)j(a+ η − 1)jj!(a+ γ + i

√x− γ2)j(a+ γ − i

√x− γ2)j

,

where γ = (1 +α+β−a− b)/2 and η = 0 when µ0 = 0 and η = 1 when µ0 = αβ �= 0. Notethat Pn(x; a, b, α, β, η) reduces to the continuous dual Hahn polynomials, see Andrews andAskey [1] and Askey and Wilson [3] when α = 0 or β = 0.

It may be hazardous to speculate who was the first one to study the associated orthogonalpolynomials or who used the adjective “associated” to describe them, but Humbert’s 1918paper [20] is the earliest work that I could find in the literature. Hahn [19] seems to bethe first to study the associated Laguerre polynomials. Although he did not find theirorthogonality relation he found the fourth order differential equation that they satisfy byfirst expressing them as sums of products of confluent hypergeometric functions. Of allthe earlier works on associated orthogonal polynomials, however, Pollaczek’s work [38–41]was probably the most important because his results constituted a significant departurefrom the classical orthogonal polynomials (belonging to the Szego class) on one hand, and ageneralization to the associated ones on the other. In its full generality Pollaczek [41] gavethe following 3-term recurrence relation

(n+ c+ 1)Pλn+1(x) = 2[(n+ c+ λ+ a)x+ b]Pλn (x) − (n+ c+ 2λ− 1)Pλn−1(x),(1.37)

with Pλ−1(x) = 0, Pλ0 (x) = 1, |x| ≤ 1, and either a > |b|, 2λ + c > 0, c ≥ 0 or a > |b|,2λ+c ≥ 1, c > −1. Pollaczek proved the orthogonality of {Pλn (x)} on [−1, 1] wrt the weight

6

function

w(λ)(x; a, b, c) =(2 sin θ)2λ−1e(2θ−π)t

2πΓ(2λ+ c)Γ(c+ 1)(1.38)

× |Γ(λ+ c+ it)|2| 2F1(1 − λ + it, c; c+ λ + it; e2iθ)|−2,

where t = (a cos θ + b)/ sinθ, x = cos θ, and derived the following expression:

Pλn (x) = Pλn (x; a, b, c)(1.39)

=A−1Bn − AnB−1

A−1B0 − B−1A0,

with

An =Γ(2λ+ c+ n)ei(c+n)θ

Γ(c+ n+ 1)Γ(2λ) 2F1(−c− n, λ+ it; 2λ; 1 − e−2iθ),(1.40)

Bn =Γ(1 − λ + it)Γ(1 − λ− it)

Γ(2 − 2λ)(2 sinθ)1−2λei(2λ+c+n−1)θ

× 2F1(1 − 2λ− c− n, 1 − λ+ it; 2 − 2λ; 1 − e−2iθ),

n = −1, 0, 1, . . ., provided 2λ is not an integer, see also [13].Taking the limit a → 0, replacing θ and t by φ and x, respectively, one obtains the

limiting relation:

(n+ c+ 1)Pλn+1(x)(1.41)

= 2[(n+ c+ λ) cosφ+ x sinφ]Pλn (x) − (n+ c+ 2λ− 1)Pλn−1(x),

with Pλ−1(x) = 0, Pλ0 (x) = 1, 0 < φ < π, 2λ + c > 0, c ≥ 0, or 0 < φ < π, 2λ + c ≥ 1,c > −1. The resulting polynomials, called by Askey and Wimp [5] the associated Meixner-Pollaczek polynomials, (Meixner [34] had also found them and their orthogonality relationwhen c = 0) are orthogonal on the infinite interval −∞ < x < ∞ wrt the same weightfunction (1.38) with appropriate replacement of the variables. If we replace λ and x in(1.41) by 1

2 (α + 1) and −x/2 sinφ, respectively, and take the limit φ → 0, then (1.41)becomes the 3-term recurrence relation for the associated Laguerre polynomials, Lαn(x; c),as was observed by Pollaczek in [40]. Askey and Wimp [5] took advantage of his propertyto obtain the weight function for the orthogonality of {Lαn(x; c)} on 0 < x <∞, namely,

Wα(x; c) =xαe−x

|Ψ(c, 1 − α; xe−πi)|2 ,(1.42)

whereΨ(a, b; x) =

1Γ(a)

∫ ∞

0e−xtta−1(1 + t)c−a−1dt, Re a > 0(1.43)

is the second solution of the confluent hypergeometric equation. Askey and Wimp [5] alsofound the weight function for the associated Hermite polynomials Hn(x; c):

w(x; c) = |D−c(ix√

2)|−2,(1.44)

7

where limb→∞ ba/2Ψ(a, b+ 1; x√b+ b) = ex

2/4D−a(x), and the expression

Hn(x; c) =�n

2�∑

k=1

(−2)k(c)k(n− k)!k!(n− 2k)!

Hn−2k(x).(1.45)

These results were derived by a different method in [22].In section 2 we shall give a brief summary of the methods of determining the spectral

measures dµ(x) and in section 3 we discuss in some details the case of the two families ofAskey-Wilson polynomials. Finally we consider some special and limiting cases in section 4.

2 Determination of the spectral measure dµ(x)

As we saw in the previous section the existence of a positive measure dµ(x) for an OPS{pn(x)} defined by the 3-term recurrence relation (1.11) or (1.21) along with the statedconditions is guaranteed by Favard’s theorem, but nothing else can be said about thismeasure without a good deal of additional work. First, one has to determine the bounds ofthe support of dµ for which one can use a characterization theorem of Wall and Wetzel [46]in terms of the so-called chain sequences to find the true interval of orthogonality, or analternate approach suggested by Chihara [9], [10]. See [21] for a brief summary of Chihara’sideas. If the support turns out to be (0,∞) or (−∞,∞) then the unboundedness of itposes the additional problem of whether or not one has a determined or an undeterminedmoment problem. For a comprehensive analysis of this problem see [44]. However, fora finite interval [a, b] the measure is necessarily unique, see [45], and can be obtained, inprinciple, in 4 different ways.

There is another approach to the associated OPS problem, mainly due to Grunbaum[15], [16], who looks at them from the point of view of the bispectral problem. Spacelimitations, however, restrict us to just refer the reader to his papers.

I. Method of moments. The moments of the measure dµ(x) are defined byµn =

∫ ba x

ndµ(x). Assuming that the measure is normalized to unity so that µ0 = 1,the first few moments can be computed by using the recurrence relation (1.11) and theorthogonality property (1.12), whether or not they are for the associated polynomials.A pattern may or may not emerge from these calculations. If it does then a guesscan be made about the general formula and, hopefully, proved by induction. Thenone can compute the function

F (z) =∞∑n=0

µnz−n−1, |z| > r,(2.1)

which is really an asymptotic expansion of F (z) as z → ∞, that converges in theexterior of any circle |z| = r containing the interval of orthogonality in its interior.

8

For z �∈ [a, b], it is well- known that

F (z) =∫ b

a

dµ(t)

z − t(2.2)

is the Stieltjes transform of dµ(t). Then the familiar Perron- Stieltjes inversion for-mula tells us that

µ(t2) − µ(t1) =1

2πilimε→0+

∫ t2

t1{F (t− iε) − F (t+ iε)}dt(2.3)

if and only if (2.2) holds, see [21], [44].

II. The generating function method. When the interval of orthogonality isbounded (or, more generally, when the moment problem is determined) one can showthat

F (z) = limn→∞

P (1)n (z)

Pn(z)=

∫ ∞

−∞

dµ(t)

z − t,(2.4)

which may be compared with (1.26), where z is in the complex plane cut along thesupport of dµ, and the convergence is uniform on compact subsets of this cut plane,see [44].

Note that it is not necessary to compute the denominator and numerator poly-nomials explicitly, only their asymptotic behaviour as n → ∞. It was pointed outin [2] that the asymptotic method of Darboux [45] can be applied to the generatingfunctions of both {Pn(x)} and {P (1)

n (x)} to determine their asymptotic behaviour.Darboux’s theorem, as stated in [36], is as follows.

Let

f(z) =∞∑

n=−∞anz

n(2.5)

be the Laurent expansion of an analytic function f(z) in an annulus 0 < |z| < r < ∞.By Cauchy’s formula

an =1

2πi

∫C

f(z)

zn+1dz,(2.6)

where C is a simple closed contour in the annulus around r = 0. Let r be thedistance from the origin of the nearest singularity of f(z), and suppose we can find a“comparison” function g(z) such that

(i) g(z) is holomorphic in 0 < |z| < r;

(ii) f(z) − g(z) is continuous in 0 < |z| < r (this can be weakened, see [36]);

(iii) the coefficients bn in the Laurent expansion

g(z) =∞∑

n=−∞bnz

n(2.7)

9

have known asymptotic behaviour. Then

an = bn + o(r−n), n → ∞.(2.8)

Since G(k)(z, t) :=∑∞n=0 P

(k)n (z)tn, the generating function of the orthogonal poly-

nomials P (k)n (z), can be computed from their 3-term recurrence relations (at least in

principle) without the explicit knowledge of the polynomials themselves, one canusually locate its singularities in the complex plane which enables one to constructsimpler “comparison” functions with known asymptotic behaviour. Therefore, thismethod is inherently simpler to use than the other methods, and have been used ina number of works on associated OPS. See, for example, [2], [22], [23], [21], [8].

III. Special function methods. These methods are usually quite computationaland rely more on inspired guesses than on the elaborate machinery of the theory ofgeneral OPS. The first step is to find an explicit form of the polynomial solution of therecurrence relation. In the case of classical orthogonal polynomials this solution alsosatisfies a second-order differential or difference equation which can be transformed,if necessary, into a self-adjoint form. The weight function usually reveals itself fromthat form. Consider, for example, the Askey-Wilson polynomials:

Pn(x; a, b, c, d|q)(2.9)

= 4φ3

[q−n, abcdqn−1, aeiθ, ae−iθ

ab, ac, ad; q, q

], n = 0, 1, . . . ,

where the 4φ3 series is a balanced and terminating basic hypergeometric series definedin [14], see also [3]. Askey and Wilson [4] had shown in an earlier paper that if oneof a, b, c, d is of the form q−n, n = 0, 1, . . ., then

φ(a, b) := 4φ3

[a, b, c, de, f, g

; q, q], efg = abcdq,(2.10)

satisfies the contiguous relation

Aφ(aq−1, bq) +Bφ(a, b) + Cφ(aq, bq−1) = 0,(2.11)

where

A = b(1− b)(aq − b)(a− e)(a− f)(a− g),

C = −a(1− a)(bq − a)(b− e)(b− f)(b− g),(2.12)

B = C − A+ ab(a− bq)(a− b)(aq− b)(1− c)(1 − d).

So, if we replace a, b by q−n and abcdqn−1, respectively, and then c, d, e, f, g byaeiθ, ae−iθ, ab, ac, ad, in that order, then we obtain the 3-term recurrence relationfor pn(x; a, b, c, d | q):

2x pn(x) = An pn+1(x) +Bn pn(x) + Cn pn−1(x),(2.13)

10

with

An =a−1(1− abqn)(1− acqn)(1− adqn)(1− abcdqn−1)

(1 − abcdq2n−1)(1− abcdq2n),

Cn =a(1− bcqn−1)(1 − bdqn−1)(1− cdqn−1)(1 − qn)

(1− abcdq2n−2)(1− abcdq2n−1),(2.14)

Bn = a + a−1 − An − Cn.

However, if we had replaced a, b, c, d, e, f, g by aeiθ, ae−iθ, q−n, abcdqn−1, ab, ac, ad,respectively, with x = cos θ, then we would obtain the following relation from (2.11)and (2.12):

q−n(1− qn)(1− abcdqn−1)rn(eiθ)(2.15)

= λ(−θ){rn(q−1eiθ) − rn(eiθ)} + λ(θ){rn(qeiθ) − rn(e

iθ)},

where rn(eiθ) := pn(cos θ; a, b, c, d|q), and

λ(θ) =(1− aeiθ)(1− beiθ)(1− ceiθ)(1− deiθ)

(1− e2iθ)(1− qe2iθ).(2.16)

Askey and Wilson [3] manipulated (2.15) to obtain the second-order divided-differenceequation in a “self-adjoint” form:

Dq

[w(x; aq1/2, bq1/2, cq1/2, dq1/2)Dqpn(x)

]+ λnw(x; a, b, c, d)pn(x) = 0(2.17)

where

w(x; a, b, c, d)dx =(e2iθ, e−2iθ; q)∞

|(aeiθ, beiθ, ceiθ, deiθ; q)∞|2dx√

1− x2(2.18)

is the absolutely continuous measure on [−1, 1].Unfortunately, however, this simple device does not seem to work for nonclassi-

cal OPS, and certainly not for the associated ones, classical or otherwise. Magnus[30] showed that the OPS belonging to what he calls the Laguerre-Hahn class, de-fined in terms of a Riccati-type difference equation, that includes the classical OPSalong with the associated ones, generally satisfy linear difference or differential equa-tions of fourth order. Magnus did not attempt to obtain the measure of orthog-onality for the associated Askey-Wilson polynomials, pαn(x; a, b, c, d|q), which satisfythe 3-term recurrence relation (2.13) with An, Bn, Cn replaced by An+α, Bn+α, Cn+α,but gave a scheme of how to derive the corresponding fourth order difference equa-tion. Wimp [49], however, was able to find an explicit fourth order differentialequation for the associated Jacobi polynomials which are the q → 1 limit cases ofpαn(x; q

1/2, qα+1/2,−qβ+1/2,−q1/2|q).Considering the complexity of the weight function wλ(x; a, b, c) in (1.37) for the

Pollaczek polynomials it would appear that guessing the measure of orthogonality

11

for the associated OPS or to derive it from special function formulas would be adaunting task indeed. Barrucand and Dickinson [6] discovered the weight function forthe associated Legendre polynomials satisfying eqn. (1.10) by a clever manipulationof the Legendre functions of both kinds, without realizing that their results follow asspecial cases of Pollaczek’s formulas given in (1.36)- -(1.39), see also [12], [38].

One may describe the special function methods and even the moment methodspretty adhoc and simple-minded, but they can be quite effective in determining themeasure of orthogonality, sometimes even when it is not unique, for example, theVn(x; a) polynomials of Al-Salam and Carlitz, see [9].

IV. Method of minimal solutions. This method, extensively used by Masson [31–33] and his collaborators [17–18], [21], [26], relies on the following ideas. A solutionX(s)n of the 3-term recurrence relation

Xn+1 = AnXn +BnXn−1, n ≥ 0,(2.19)

is a minimal (or subdominant) solution if for any other linearly independent solutionX(d)n (dominant) one has the property

limn→∞

X(s)n

X(d)n

= 0.(2.20)

Pincherle’s theorem (see [31] and the references therein) states that (2.20) is a nec-essary and sufficient condition for the convergence of the continued fraction thatcorresponds to (2.19), namely,

B0

A0+

B1

A1+

B2

A2+· · · .(2.21)

In case of convergence the limit is simply given by −X(s)0

X(s)−1

, assuming, of course, that

Bn �= 0, n ≥ 0. When a minimal solution exists it is unique up to a multiple indepen-dent of n. However, it was noted in [33] by means of an example that the minimalsolution may change in different parts of the complex plane or from n ≥ 0 to n ≤ 0.An example of how to construct a minimal solution, when it exists, by first derivinga set of solutions of (2.19) (any two of them being linearly independent) is given in [17].

3 Associated Askey-Wilson polynomials

The associated polynomials that generalize the Askey-Wilson polynomials given in(2.9) are solutions of the 3-term recurrence relation

(z + z−1 − a− a−1 + An+α + Cn+α)pαn(x)(3.1)

12

= An+αpαn+1(x) + Cn+αp

αn−1(x), n = 0, 1, 2, . . . ,

with pα−1(x) = 0, pα0 (x) = 1; An+α, Cn+α being the same as in (2.14) with n replacedby n + α. It was shown in [27] that the two linearly independent solutions of (3.1)are

rn+α =(abqn+α, acqn+α, adqn+α, bcdqn+α/z; q)∞

(bcqn+α, bdqn+α, cdqn+α, azqn+α; q)∞

(a

z

)n+α

(3.2)

× 8W7(bcd/qz; b/z, c/z, d/z, abcdqn+α−1, q−α−n; q, qz/a)

and

sα+n =(abcdq2n+2α, bzqα+n+1, czqn+α+1, dzqn+α+1, bcdzqn+α; q)∞

(bcqn+α, bdqn+α, cdqn+α, qn+α+1, bcdzq2n+2α+1; q)∞(az)n+α(3.3)

× 8W7(bcdzq2n+2α; bcqn+α, bdqn+α, cdqn+α, qn+α+1, zq/a; q, az),

where

2r+1W2r(a; a1, a2, . . . , a2r−2; q, z)(3.4)

:= 2r+1φ2r

[a, qa1/2, − qa1/2, a1, . . . , a2r−2

a1/2, − a1/2, qa/a1, . . . , qa/a2r−2; q, z

],

is a very-well-poised basic hypergeometric series, see [14] for definitions, propertiesand relevant formulas. Taking a linear combination of these solutions along with thegiven initial conditions (pα−1 = 0, pα0 = 1) leads to the following expression:

pαn(x) =(bc, bd, cd, bcqα−1, bdqα−1, cdqα−1, qα, az; q)∞ za1−2α

(1− abcdq2α−2)(abqα, acqα, adqα, abcdqα−1, bz, cz, dz, bcd/z; q)∞(3.5)

× {sα−1(z)rn+α(z) − rα−1(z)sn+α(z)},

where 2x = z + z−1, from which one gets the asymptotic formula:

limn→∞

(z/a)npαn(x) =(az)1−α(az, bcqα−1, bdqα−1, cdqα−1, qα; q)∞sα−1(z)

(1− abcdq2α−2)(abqα, acqα, adqα, abcdqα−1, z2; q)∞(3.6)

for |z| < 1, i.e. x ∈ C/[−1, 1], and |a| < 1.A second OPS satisfying (3.1) but a different initial condition

qα0 (x) = 1, qα1 (x) = 1 + A−1α (z + z−1 − a− a−1),(3.7)

which corresponds to the case of zero initial death rate in the birth- and-death modeldiscussed in section 1, was found to be, see [27]:

qαn(x) =(bc, bd, cd, bcqα−1, bdqα−1, cdqα−1, qα, az; q)∞za

1−2α

(1− abcdq2α−2)(abqα, acqα, adqα, abcdqα−1, bz, cz, dz, bcd/z; q)∞(3.8)

× {(sα−1(z) − sα(z))rn+α(z) − (rα−1(z) − rα(z))sn+α(z)},

13

and hence, for |z| < 1,

limn→∞

(z/a)nqαn(x) =(az)1−α(bcqα−1, bdqα−1, cdqα−1, qα, az; q)∞(az)1−α

(1 − abcdq2α−2)(abqα, acqα, adqα, abcdqα−1, z2; q)∞(3.9)

× (sα−1(z) − sα(z)).

Let us denote the measures of orthogonality for {pαn(x)} and {qαn(x)} by dµ(1)(x ;α)and dµ(2)(x ;α), respectively, with∫ ∞

−∞dµ(i)(x ;α) = 1 ,

∫ ∞

−∞pn(x)pm(x)dµ(1)(x ;α) = π(1)

n δmn,(3.10) ∫ ∞

−∞qn(x)qm(x)dµ(2)(x;α) = π(2)

n δm,n,

i = 1, 2.Since An+α, Bn+α and Cn+α of (3.1) and (2.14) are bounded if |a|, |b|, |c|, |d| < 1

and α ≥ 0 (we shall assume these restrictions to hold) the supports of both dµ(i)(t;α),i = 1, 2, are bounded, by [9, Th. IV 2.2] and consequently the moment problem isdetermined. Since the numerator polynomial corresponding to both pαn(x) and qαn(x)is a multiple of pα+1

n−1(x) (which can be easily proved) formula (2.4) now takes the form

∫ ∞

−∞

dµ(1)(t;α)

x− t=

2z(1 − bcdzq2α−1)(1 − bcdzq2α)

(1 − bzqα)(1 − czqα)(1 − dzqα)(1− bcdzqα−1)(3.11)

× 8W7(bcdzq2α; bcqα, bdqα, cdqα, qα+1, zq/a; q, az)

8W7(bcdzq2α−2; bcqα−1, bdqα−1, cdqα−1, qα, zq/a; q, az),

and

∫ ∞

−∞

dµ(2)(t;α)

x− t=

2z(1 − bcdzq2α−1)(1 − bcdzq2α)

(1 − bzqα)(1− czqα)(1 − dzqα)(1− bcdzqα−1)(3.12)

× 8W7(bcdzq2α; bcqα, bdqα, cdqα, qα+1, zq/a; q, az)

8W7(bcdzq2α−2; bcqα−1, bdqα−1, cdqα−1, qα, z/a; q, aqz),

for x ∈ C/[−1, 1]. Using a very useful theorem of Nevai [35, Corollary 36, p. 141,Theorem 40, p. 143] one can show that the support of the absolutely continuous partof the measure is (−1, 1) and that the jump function j(x) for the discrete part dj(x)is constant in (−1, 1). Furthermore it was shown in [27] that there are no discretemasses outside (−1, 1) or at the points x = ±1. So from (2.3) one deduces that

dµ(1)(cos θ;α)

dθ(3.13)

=(abqα, acqα, adqα, bcqα, bdqα, cdqα, qα+1; q)∞2π(1− abcdqα−2)(abcdqα−2, abcdq2α; q)∞

(1− abcdq2α−2)

14

×(abcdq2α−2; q)∞

∣∣∣∣ (e2iθ, qα+1eeiθ; q)∞(aqαeiθ, bqαeiθ, cqαeiθ, dqαeiθ, qe2iθ; q)∞

∣∣∣∣2

×∣∣∣ 8W7(q

αe2iθ; qeiθ/a, qeiθ/b, qeiθ/c, qeiθ/d, qα; q, abcdqα−2)∣∣∣−2

.

It was also shown in [27] that dµ(2)(x;α) has at most one discrete mass for 0 <q < 1, α ≥ 0 and −1 < a, b, c, d < 1, and that

dµ(2)(cos θ;α)

dθ(3.14)

=(abqα, acqα, adqα, bcqα, bdqα, cdqα, qα+1; q)∞

2π(abcdq2α; q)∞

×(abcdq2α−1; q)∞(abcdqα−1; q)∞

1− 2axqα + a2q2α

1− 2ax+ a2

× |(e2iθ, qα+1e2eiθ; q)∞|2

|(aqαeiθ, bqαeiθ, cqαeiθ, dqαeiθ, qe2iθ; q)∞|2

×∣∣∣ 8W7(q

αe2iθ; eiθ/a, qeiθ/b, qeiθ/c, qeiθ/d, qα; q, abcdqα−1)∣∣∣−2

.

Explicit forms of pαn(x) and qαn(x) found in [27] are:

pαn(x) =n∑k=0

(q−n, abcdq2α+n−1, abcdq2α−1, aeiθ, ae−iθ; q)k(q, abqα, acqα, adqα, abcdqα−1; q)k

qk(3.15)

× 10W9

(abcdq2α+k−2; qα, bcqα−1, bdqα−1, cdqα−1, qk+1,

abcdq2α+n+k−1, qk−n; q, a2),

and

qαn(x) =n∑k=0

(q−n, abcdq2α+n−1, abcdq2α−1, aeiθ, ae−iθ; q)k(q, abqα, acqα, adqα, abcdqα−1; q)k

qk(3.16)

× 10W9

(abcdq2α+k−2; qα, bcqα−1, bdqα−1, cdqα−1, qk,

abcdq2α+n+k−1, qk−n; q, qa2).

Masson’s exceptional Askey-Wilson polynomials, see [32], [25] that correspond to theindeterminate cases abcd = q or q2 (see eqn. (2.14)) turn out to be the limiting caseα → 0+ of pαn(x), while qαn(x) approaches the Askey-Wilson polynomials.

By using the transformation theory of basic hypergeometric series a simpler formof (3.14) was found in [42]

15

pαn(x) ≡ pαn(x; a, b, c, d|q)(3.17)

=(abcdq2α−1, qα+1; q)n

(q, abcdqα−1; q)nq−αn

n∑k=0

(q−n, abcdq2α+n−1; q)k(qα+1, abqα; q)k

×(aqαeiθ, aqαe−iθ; q)k(acqα, acqα; q)k

qkk∑j=0

(qα, abqα−1, acqα−1, adqα−1; q)j(q, abcdq2α−2, aqαeiθ, aqαe−iθ; q)j

qj.

Assuming that max(|a|, |b|, |c|, |d|) < q(1−α)/2, α > 0, one can derive the followingintegral representation [42]:

pαn(x; a, b, c, d|q)(3.18)

=∫ 1

−1K(x, y)

(abcdq2α−1, qα+1; q)n(q, abcdqα−1; q)n

q−αnpn(y; aqα/2, bqα/2, cqα/2, dqα/2|q)dy,

where

K(x, y) =(q, q, q, abqα−1, acqα−1, adqα−1, bcqα−1, bdqα−1, cdqα−1, qα; q)∞

4π2(abcdq2α−2, qα+1; q)∞√

1 − y2(3.19)

× (e2iφ, e−2iφ; q)∞h(y; qα/2eiθ, qα/2e−iθ)

∫ π

0

(e2iψ, e−2iψ; q)∞

h(cosψ; aqα−12 , bq

α−12 , cq

α−12 , dq

α−12 )

×h(cosψ; qα+1

2 eiθ, qα+1

2 e−iθ

h(cosψ; q1/2eiφ, q1/2e−iθ)dψ,

where y = cosφ, 0 ≤ φ ≤ π, and

h(cos θ; a1, a2, . . . , ak) :=k∏j=1

h(cos θ; aj)

h(cos θ; a) = (aeiθ, ae−iθ; q)∞ =∞∏k=0

(1− 2aqk cos θ + a2q2k).

(3.20)

This representation enables us to compute many formulas for the associated polyno-mials pαn(x; a, b, c, d|q) from the corresponding formulas for the ordinary Askey-Wilsonpolynomials pn(x; aq

α/2, bqα/2, cqα/2, dqα/2|q), see, for example, [43].By replacing a, b, c, d and α in (3.17) by q1/2, qα+1/2, −qβ+1/2, −q1/2 and c,

respectively, then taking the limit q → 1 and defining

P (α,β)n (x; c)(3.21)

=(α+ c+ 1)n(α + β + c+ 1)n(c+ 1)n(α + β + 2c + 1)n

limq→1

pcn(x; q1/2, qα+1/2,−qβ+1/2,−q1/2 | q

),

16

one can show that

P (α,β)n (x; c) =

(α+ c+ 1)nn!

n∑k=0

(−n, α+ β + 2c + n+ 1)k(c+ 1, α + c+ 1)k

(1 − x

2

)k(3.22)

×k∑j=0

(−k, c, α+ c)jj! (−k, α+ β + 2c)j

( 2

1 − x

)j

=cΓ(α+ β + 2c)

Γ(α+ c)Γ(β + c)21−α−β−3c

×∫ 1

−1

∫ 1

−1dydz(1 − y)α+c−1(1 + y)β+c−1(1− z)c−1

× P (α+c,β+c)n

(x(1 + z) + y(1− z)

2

),

Re(α+ c) > 0, Re(β + c) > 0, Re c > 0.

4 Special and Limiting cases

I. Continuous q-Jacobi polynomials. Let us take

a = q1/2, b = qβ+1/2, c = −qα+1/2, d = −q1/2(4.1)

and replace α by c in (3.14). The 10φ9 then is a balanced, terminating and very-well-poised series which can be transformed by [14, III. 28]. Thus

10W9

(qα+β+2c+k; qc,−qα+β+c,−qβ+c, qα+c, qk+1, qα+β+2c+n+k+1, qk−n; q, q

)(4.2)

=(qc+1, qα+β+2c+1,−q,−qα+β+c+1; q)n(q

α+β+c+1,−qc+1, q,−qα+β+2c+1; q)k( − qc+1,−qα+β+2c+1, q, qα+β+c+1; q)n( − qα+β+c+1, qc+1,−q, qα+β+2c+1; q)k

× 10W9

(− qα+β+2c+k; qc,−qα+β+c, qβ+c,−qα+c,−qk+1, qα+β+2c+n+k+1, qk−n; q, q

).

Hence the explicit form of the associated continuous q-Jacobi polynomials is:

pcn(x; q1/2, qβ+1/2,−qα+1/2,−q1/2|q)(4.3)

=(qc+1, qα+β+2c+1,−q,−qα+β+c+1; q)n(q, qα+β+c+1,−qc+1,−qα+β+2c+1; q)n

n∑k=0

(q−n, qα+β+2c+n+1,−qα+β+2c+1; q)k(qc+1, qβ+c+1,−qα+c+1; q)k

×(q1/2eiθ, q1/2e−iθ; q)k(−q,−qα+β+c+1; q)k

qk

× 10W9

(− qα+β+2c+k; qc,−qα+β+c, qβ+c,−qα+c,−qk+1,

qα+β+2c+n+k+1, qk−n; q, q)

17

As q → 1 this approaches a multiple of Wimp’s [49, eqn. (19)] expression for theassociated Jacobi polynomials:

(c+ 1)n(α+ β + 2c+ 1)nn! (α + β + c+ 1)n

n∑k=0

(−n)k(α+ β + 2c+ n+ 1)k(c+ 1)k(β + c+ 1)k

(1− x

2

)k(4.4)

× 4F3

[k − n, α+ β + 2c + n+ k + 1, c, β + cc+ k + 1, β + c+ k + 1, α+ β + 2c

; 1].

II. Continuous q-ultraspherical polynomials. Instead of (4.1) let us now setb = aq1/2, c = −a, d = −aq1/2 and denote

pαn(x; a, aq1/2,−a− aq1/2|q) =

(qα+1; q)n(a4qα; q)n

anCαn (x; a

2|q)(4.5)

where Cαn (x; a

2|q) is the associated q-ultraspherical polynomials which reduces to theq-ultraspherical polynomials

Cn(x; a2|q) =

n∑k=0

(a2; q)k(a2; q)n−k

(q; q)k(q; q)n−kei(n−2k)θ,(4.6)

when α = 0.Use of the transformation formula [14, (3.4.7)] and some simplifications gives

Cαn (x; a

2|q) =(a4qα−1; q)∞(qα+1; q)∞

zn

1 − z−2 2φ1

[q/a2z2, q/a2

q/z2 ; q, a4qα−1]

(4.7)

× 2φ1

[qz2/a2, q/a2

qz2 ; q, a4qα+n]+ (z ↔ z−1),

where z = eiθ, x = cos θ, 0 < θ < π. From this the generating function followsimmediately:

Gαt (x; β|q) =

∞∑n=0

Cαn (x; β|q)tn(4.8)

=(β2qα−1; q)∞

(qα+1; q)∞(1− zt)(1− z−2)2φ1

[q/β, q/βz2

q/z2 ; q, β2qα−1]

× 3φ2

[q/β, qz2/β, zt

qz2, qzt; q, β2qα

]+ (z ↔ z−1),

|t| < 1. We can now combine the two terms on the rhs of (4.8) in the following way.First, note that

Gαt (x; β|q) =

(β2qα−1; q)∞(qα+1; q)∞(1− z−2)

∞∑j=0

∞∑k=0

(q/β, q/βz2; q)j(q, q/z2; q)j

(4.9)

18

×(q/β, qz2/β; q)k(q, qz2; q)k

(β2qα)j+kq−j(1− z−2qj−k)

(1− ztqk)(1 − tqj/z).

Setting k = n− j we find, after some simplifications, that

Gαt (x; β|q) =

(β2qα−1; q)∞(qα+1; q)∞(1 − zt)(1− t/z)

∞∑n=0

(q/β, qz2/β, zt; q)n(q, z2, qzt; q)

(β2qα−1)n(4.10)

× 8W7(q−n/z2; q/β, q/βz2, t/z, q−n/zt, q−n; q, β2).

However, using [14, III. 16] and then [14, III.15] we get

8W7(q−n/z2; q/β, q/βz2, t/z, q−n/zt, q−n; q, β2)(4.11)

=(z2, q/β2, qzt; q)n(qz2/β, q/β, zt; q)n

4φ3

[q−n, βzt, βt/z, qqzt, qt/z, q−n/β2 ; q, q

].

Hence

Gαt (x; β q) =

(β2qα−1; q)∞(qα+1; q)∞(1 − zt)(1− t/z)

∞∑n=0

(q/β2; q)n(q; q)n

(β2qα−1)n(4.12)

× 4φ3

[q−n, βzt, βz/z, qqzt, qt/z, q−n/β2 ; q, q

]

=1− qα

(1 − 2xt+ t2)3φ2

[βteiθ, βte−iθ, q

qteiθ, qte−iθ; q, qα

],

which is the same as [8, eqn. (2.8)].

III. Associated Wilson polynomials. Wilson polynomials Pn(x; a, b, c, d) satisfythe 3-term recurrence relation

(λn + µn − x)Pn(x) = λnPn+1(x) + µnPn−1(x),(4.13)

P−1(x) = 0, P0(x) = 1, x = a2 + t2,

where

λn =(n+ a + b)(n+ a+ c)(n + a+ d)(n + s− 1)

(2n + s− 1)(2n + s),

µn =n(n+ b+ c− 1)(n + b+ d − 1)(n+ c+ d − 1)

(2n+ s− 2)(2n + s− 1),(4.14)

s = a + b+ c+ d,

see [47], [48] and [3].

19

Wilson [47] found that they are orthogonal on (−∞,∞) wrt the weight function∣∣∣∣∣Γ(a + it)Γ(b+ it)Γ(c+ it)Γ(d+ it)

Γ(2it)

∣∣∣∣∣2

and that

Pn(x; a, b, c, d) = 4F3

[ −n, n+ s− 1, a− it, a+ ita + b, a+ c, a + d

; 1].(4.15)

The associated Wilson polynomials are, of course, the solutions of (4.13) with nreplaced by n+α in λn and µn. Their weight function and the orthogonality propertieswere worked out by Ismail et al. [26], see also [33] and [24]. In fact, the authors in [26]found two families of polynomials that correspond to the two cases we have consideredin the previous section. Their weight function [26, (3.39)] for the first family can befound as the q → 1− limit of our formula (3.12) after having replaced a, b, c, d, α,abcd,eiθ by qa, qb, qc, qd, qγ and qiτ , respectively. Similarly, formula [26, (4.25)] follow asthe q → 1− limit of (3.13) with the same replacements. Furthermore, the functionalforms for the two polynomial systems are given by

P γn (x; a, b, c, d) =

n∑k=0

(−n, s+ 2γ + n − 1, s + 2γ − 1, a + iτ, a− iτ )kk!(a+ b+ γ, a+ c + γ, a+ d + γ, s+ γ − 1)k

(4.16)

9F8

(s+ 2γ + k − 2; γ, b+ c+ γ − 1, b + d+ γ − 1,

c+ d+ γ − 1, k + 1, s+ 2γ + n + k − 1, k − n; 1)

as the q → 1− limit of (3.14), and

Qγn(x; a, b, c, d)(4.17)

=n∑k=0

(−n, s+ 2γ + n− 1, s+ 2γ − 1, a + iτ, a− iτ )kk!(a + b+ γ, a+ c+ γ, a+ d+ γ, s+ γ − 1)k

× 9F8(s+ 2γ + k − 2; γ, b+ c+ γ − 1, b+ d+ γ − 1, c+ d + γ − 1, k,

s+ 2γ + n + k − 1, k − n; 1),

as the q → 1− limit of (3.15), where

9F8(a; b, c, d, e, f, g, h, )(4.18)

:=∞∑n=0

a + 2n

a

(a, b, c, d, e, f, g, h)nn! (1 + a− b, 1 + a− c, 1 + a− d, 1 + a− e, 1 + a− f, 1 + a− g, 1 + a− h)n

.

Acknowledgement. I wish to express my thanks and appreciation to Mourad Ismailfor his help and suggestions in preparing the final version of this paper.

20

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