the diagonal general case of the laguerre-sobolev type
TRANSCRIPT
Revista Colombiana de MatematicasVolumen 47(2013)1, paginas 39-66
The Diagonal General Case of the
Laguerre-Sobolev Type Orthogonal
Polynomials
El caso general diagonal de los polinomios ortogonales de tipoLaguerre-Sobolev
Herbert Duenas1,B, Juan C. Garcıa1, Luis E. Garza2,Alejandro Ramırez2
1Universidad Nacional de Colombia, Bogota, Colombia
2Universidad de Colima, Colima, Mexico
Abstract. We consider the family of polynomials orthogonal with respect tothe Sobolev type inner product corresponding to the diagonal general case ofthe Laguerre-Sobolev type orthogonal polynomials. We analyze some proper-ties of these polynomials, such as the holonomic equation that they satisfyand, as an application, an electrostatic interpretation of their zeros. We alsoobtain a representation of such polynomials as a hypergeometric function, andstudy the behavior of their zeros.
Key words and phrases. Orthogonal polynomials, Laguerre-Sobolev type poly-nomials, Laguerre Polynomials, Derivative of a Dirac Delta.
2010 Mathematics Subject Classification. 33C47, 42C05.
Resumen. Se considera la familia de los polinomios ortogonales con respectoa un producto interno de tipo Sobolev correspondiente al caso general di-agonal de los polinomios ortogonales de tipo Laguerre-Sobolev. Se analizanalgunas propiedades de estos polinomios tales como la ecuacion holonomicaque satisfacen y, como una aplicacion de dicha ecuacion, una interpretacionelectrostatica de sus ceros. Tambien se obtiene una representacion de talespolinomios en terminos de una funcion hipergeometrica, y se estudia el com-portamiento de sus ceros.
Palabras y frases clave. Polinomios ortogonales, polinomios de tipo Laguerre-Sobolev, polinomios de Laguerre, derivada de una Delta de Dirac.
39
40 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
1. Introduction
The Laguerre polynomials are defined as the orthogonal polynomials with re-spect to the inner product
〈p, q〉α =
∫I
p(x)q(x) dσ(x), (1)
where dσ(x) = xαe−xdx, I is the positive real line, α > −1, and p, q arepolynomials with real coefficients.
We denote by{Lαn(x)
}∞n=0
the sequence of monic Laguerre polynomials,
and by{Lαn(x)
}∞n=0
the sequence of orthogonal polynomials with respect tothe inner product
〈p, q〉S,α =
∫I
p(x)q(x) dσ +
j∑i=0
Mip(i)(0)q(i)(0), (2)
where Mi ∈ R+ and j ∈ N. The goal of this paper is to generalize some resultsobtained in [4], [5] and [10], extending them for an arbitrary finite number ofmasses.
The structure of the manuscript is as follows. In Section 2, we present somepreliminary results about the Laguerre polynomials. In Section 3, we give aconnection formula which expresses the Laguerre-Sobolev type polynomials interms of some Laguerre polynomials. This formula will be our principal tool.
A 2j + 2 terms recurrence relation for the Laguerre-Sobolev type polyno-mials is deduced in Section 4. In Section 5, we show a second order differentialequation that the Laguerre-Sobolev type polynomials satisfy. This holonomicequation will be used in Section 6 in order to find an electrostatic interpretationof the zeros of Lαn(x).
In section 7, we use the hypergeometric representation of the Laguerre poly-nomials to construct a hypergeometric representation for the Laguerre-Sobolevtype polynomials and finally, in Section 8, we analyze the location of the zerosof Lαn(x). Some numerical experiments using MatLab and Mathematica softwareare presented.
2. Preliminaries
Let µ be a linear moment functional defined in P, the linear space of polynomialswith real coefficients, and define their corresponding moments by µn = 〈µ, xn〉,n ≥ 0. If {Pn(x)}n≥0 is a polynomials sequence such that the degree of Pn(x)is n and it satisfies ⟨
µ, Pn(x)Pm(x)⟩
= Knδm,n,
with Kn 6= 0, n,m ∈ N, and δm,n is the Kronecker function defined by
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 41
δm,n =
{0, if m 6= n;
1, if n = m,
then we say that{Pn(x)
}n≥0
is the sequence of orthogonal polynomials asso-
ciated with the linear functional µ. We will assume that µ is a positive definitelinear functional, i.e.
⟨µ, π(x)
⟩> 0 for any polynomial π(x) > 0. In such a
case, µ has an integral representation in terms of a positive measure, and theorthogonality relation can be expressed in terms of an inner product such as (1).
It is well known that {Pn(x)}n≥0 satisfies the following three term recur-rence relation:
xPn(x) = Pn+1(x) + βnPn(x) + γPn−1(x),
where P−1(x) = 0, P0(x) = 1 and, for n ∈ N, the recurrence coefficients{βn}n∈N, {γn}n∈N are real with γn 6= 0. The n-th reproducing Kernel Kn(x, y)associated with
{Pn(x)
}n≥0
is defined by
Kn(x, y) =
n∑k=0
Pk(x)Pk(y)
‖Pk‖2µ,
where ‖Pk‖2µ =⟨µ,(P (x)
)2⟩.
There is an explicit expression forKn(x, y), the so-called Christoffel-Darbouxformula, given by
Kn(x, y) =Pn+1(x)Pn(y)− Pn(x)Pn+1(y)
(x− y)‖Pn‖2µ,
for x 6= y. We will use the following notation for the partial derivatives ofKn(x, y)
∂i+t(Kn(x, y)
)∂xi∂yt
= K(i,t)n (x, y).
If p ∈ P and deg(p) ≤ n then we have⟨K(0,i)n (x, y), p(x)
⟩= p(i)(y), (3)
which is called the reproducing property.
In [4], the Leibniz formula for the j-th derivative of a product is used todeduce the following formula:
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42 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
K(0,j)n−1 (x, a) =
j!
‖Pn−1‖2(x− a)j+1×(
Pn(a) + P (1)n (a)(x− a) +
P(2)n (a)
2!(x− a)2 + · · ·+ P
(n)n (a)
n!(x− a)n
)×(
Pn−1(a) + P(1)n−1(a)(x− a) +
P(2)n−1(a)
2!(x− a)2 + · · ·+
P(j)n−1(a)
j!(x− a)j
)−
j!
‖Pn−1‖2(x− a)j+1×(
Pn−1(a) + P(1)n−1(a)(x− a) +
P(2)n−1(a)
2!(x− a)2 + · · ·+
P(n−1)n−1 (a)
(n− 1)!(x− a)n−1
)×(Pn(a) + P (1)
n (a)(x− a) +P
(2)n (a)
2!(x− a)2 + · · ·+ P
(j)n (a)
j!(x− a)j
). (4)
In the following section, we will deduce a formula for K(i,j)n−1 (a, a), which will
help us to deduce the derivative of the polynomial Lαn(x).
The families of orthogonal polynomials that have been most extensivelystudied in the literature are the Jacobi, Laguerre and Hermite polynomials.They are the so-called classical orthogonal polynomials, and have (amongothers) the following characterization:{
Pn(x)}n∈N is classical if and only if Pn, n ∈ N, is a solution of the second
order differential equation
φ(x)y′′(x) + τ(x)y′(x) + λny(x) = 0,
where φ and τ are polynomials independent of n, with degree at most 2 andexactly 1, respectively, and λn is a constant.
In this paper, we will focus our attention on the Laguerre monic orthogonalpolynomials. In the following proposition we list some of their properties, thatwill be useful in the upcoming sections (see [2], [3] and [12]).
Proposition 1. Let {Lαn}n≥0 be the sequence of Laguerre monic orthogonalpolynomials. Then,
1) For n ∈ N,
xLαn(x) = Lαn+1(x) + (2n+ α+ 1)Lαn(x) + n(n+ α)Lαn−1(x). (5)
2) For n ∈ N,
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 43
Lαn(x) = (−1)n(α+ 1)n
∞∑k=0
(−n)k(α+ 1)k
xk
k!=
(−1)n(α+ 1)n × 1F1(−n;α+ 1), (6)
where
(a)m :=
1, if m = 0;|m−1|∏k=0
(a+ k sgn(m)), if m ∈ Z r {0}
and
sgn(m) :=
{1, if m ∈ Z+;
−1, if m ∈ Z−.
In other words, the Laguerre polynomials can be expressed as a hypergeo-metric function of the form 1F1.
3) For n ∈ N,
(Lαn)(i)(0) = (−1)n+i n!Γ(α+ n+ 1)
(n− i)!Γ(α+ i+ 1). (7)
4) For n ∈ N,‖Lαn‖2α = n!Γ(n+ α+ 1). (8)
5) For n ∈ N, Lαn(x) satisfies the differential equation
xy′′ + (α+ 1− x)y′ = −ny. (9)
6) For n ∈ N,
x(Lαn(x)
)′= nLαn(x) + n(n+ α)Lαn−1(x). (10)
7) For n ∈ N,
Lαn(x) =n!
(−1)n
n∑k=0
Γ(n+ α+ 1)
(n− k)!Γ(α+ k + 1)
(−x)k
k!. (11)
8) For n ∈ N,Lαn(x) = Lα+1
n (x) + nLα+1n−1(x). (12)
9) For n ∈ N, (Lαn)′
(x) = nLα+1n−1(x). (13)
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44 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
3. Connection Formula
Let {Pn}n≥0 be the sequence of monic orthogonal polynomials with respect
to a moment linear functional µ, and denote by{Pn(x)
}n≥0
the sequence of
polynomials orthogonal with respect to the inner product
〈p, q〉S = 〈p, q〉µ +
j∑i=0
Mi p(i)(0) q(i)(0).
Since {Pn}n≥0 is a basis in the linear space of polynomials, we can write
Pn(x) =
n∑k=0
an,kPk(x) = Pn(x) +
n−1∑k=0
an,kPk(x), (14)
where
an,k =
⟨Pn(x), Pk(x)
⟩µ
‖Pk(x)‖2µ.
Taking into account that for k < n
⟨Pn(x), Pk(x)
⟩µ
= −j∑i=0
MiP(i)n (a)P
(i)k (a),
Pn(x) can be written as
Pn(x) = Pn(x)−j∑i=0
MiP(i)n (a)
n−1∑k=0
P(i)k (a)Pk(x)
‖Pk(x)‖2µ(15)
= Pn(x)−j∑i=0
MiP(i)n (a)K
(0,i)n−1 (x, a). (16)
In order to find P(i)n (a) for i = 0, . . . , j, we need to solve the linear system,
1 +M0K
(0,0) M1K(0,1) · · · MjK
(0,j)
M0K(1,0) 1 +M1K
(1,1) · · · MjK(1,j)
M0K(2,0) M1K
(2,1) · · · MjK(2,j)
......
. . ....
M0K(j,0) M1K
(j,1) · · · 1 +MjK(j,j)
P
(0)n (a)
P(1)n (a)
P(2)n (a)
...
P(j)n (a)
=
P
(0)n (a)
P(1)n (a)
P(2)n (a)
...
P(j)n (a)
(17)
which is obtained by taking derivatives in (16), where K(p,q) := K(p,q)n−1 (a, a).
According to (4) we have
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 45
K(0,j)n−1 (x, a) =
j!
‖Pn−1‖2×
[n∑
r=j+1
P(r)n (a)
r!
(Pn−1(a)(x− a)r−j−1 + · · ·+
P(k)n−1(a)
k!(x− a)k+r−j−1 + · · ·+
P(j)n−1(a)
j!(x− a)r−1
)−
n−1∑k=j+1
P(k)n−1(a)
k!
(Pn(a)(x− a)k−j−1 + · · ·+
P(r)n (a)
r!(x− a)k+r−j−1 + · · ·+ P
(j)n (a)
j!(x− a)k−1
)]. (18)
Having in mind that we are interested in finding K(i,j)n−1 (a, a), we calculate
the i-th derivative in (18) with respect to x and evaluate it at x = a to obtain
K(i;j)n−1(a, a) =
j!
‖Pn−1‖2
[n∑
r=j+1
i!P
(r)n (a)P
(i+j+1−r)n−1 (a)
r!(i+ j + 1− r)!−
n−1∑k=j+1
i!P
(i+j+1−k)n (a)P
(k)n−1(a)
k!(i+ j + 1− k)!
]. (19)
Our goal is to write the polynomials Lαn(x), orthogonal with respect to theinner product (2), in terms of some Laguerre orthogonal polynomials. First, weneed the following theorem.
Theorem 2. For t, n ∈ N and α > −1, the Laguerre polynomials satisfy thefollowing property ([9])
Lαn(x) =
t∑k=0
(t
k
)(n)−kL
α+tn−k(x). (20)
Let{Lα+j+1n (x)
}be the sequence of Laguerre polynomials orthogonal with
respect to dσ(x) = e−xxα+j+1 dx. We will express K(0,i)n−1 (x, 0) as a linear com-
bination of Lα+j+1n (x). Then,
‖Lαn−1‖2α(Lαn−1)(i)(0)
K(0,i)n−1 (x, 0) = Lα+j+1
n−1 (x) +
n−2∑k=0
bn−1,kLα+j+1k (x), (21)
where
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46 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
bn−1,k =
⟨‖Lαn−1‖
2α
(Lαn−1)(i)(0)K
(0,i)n−1 (x, 0), Lα+j+1
k (x)
⟩α+j+1
‖Lα+j+1k (x)‖2α+j+1
=‖Lαn−1‖2α
(Lαn−1)(i)(0)‖Lα+j+1k (x)‖2α+j+1
⟨K
(0,i)n−1 (x, 0), Lα+j+1
k (x)⟩α+j+1
.
From (1) and (3), we see that for 0 ≤ k ≤ n− 2− j,
⟨K
(0,i)n−1 (x, 0), Lα+j+1
k (x)⟩α+j+1
=
∫ ∞0
K(0,i)n−1 (x, 0)Lα+j+1
k (x)xα+j+1e−x dx
=
∫ ∞0
K(0,i)n−1 (x, 0)(xj+1Lα+j+1
k (x))xαe−x dx
=⟨K
(0,i)n−1 (x, 0), xj+1Lα+j+1
k (x)⟩α
=(xj+1Lα+j+1
k (x))(i)
(0) = 0.
In other words,
K(0,i)n−1 (x, 0) =
(Lαn−1)(i)(0)
‖Lαn−1(x)‖2αLα+j+1n−1 (x)+
n−2∑k=n−j−1
⟨K
(0,i)n−1 (x, 0)Lα+j+1
k (x)⟩α+j+1
‖Lα+j+1k (x)‖2α+j+1
Lα+j+1k (x). (22)
Taking into account that
K(0,i)n−1 (x, 0) =
n−1∑m=0
Lαm(x) (Lαm)(i)
(0)
‖Lαm(x)‖2α, (23)
and using (7), (8) and (20), we have
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 47
⟨K
(0,i)n−1 (x, 0), Lα+j+1
k (x)⟩α+j+1
=
n−1∑m=0
(Lαm)(i)(0)
‖Lαm(x)‖2α
⟨Lαm(x), Lα+j+1
k (x)⟩α+j+1
=
n−1∑m=0
(Lαm)(i)(0)
‖Lαm(x)‖2α
j+1∑t=0
(j + 1
t
)(m)(−t)
⟨Lα+j+1m−t (x), Lα+j+1
k (x)⟩α+j+1
=
n−1∑m=0
j+1∑t=0
(j + 1
t
)(m)(−t)(L
αm)(i)(0)
‖Lαm(x)‖2α
⟨Lα+j+1m−t (x), Lα+j+1
k (x)⟩α+j+1
=
n−1∑m=i
j+1∑t=0
(j + 1
t
)(−1)m+im!
(m− t)!Γ(α+ i+ 1)(m− i)!‖Lα+j+1
k (x)‖2α+j+1 δk,m−t.
For n− j − 1 ≤ k ≤ n− 2, let
c(k;i) =
⟨K
(0,i)n−1 (x, 0), Lα+j+1
k (x)⟩α+j+1
‖Lα+j+1k (x)‖2α+j+1
and c(n−1;i) =(Ln−1,
α )(i)(0)
‖Lαn−1(x)‖2α,
or, equivalently,
c(k;i) =
n−1∑m=i
j+1∑t=0
(j + 1
t
)(−1)m+im!
(m− t)!Γ(α+ i+ 1)(m− i)!δk,m−t
and
c(n−1;i) =(−1)n+i−1
(n− i− 1)!Γ(α+ i+ 1). (24)
Then,
Pn(x) = Lαn(x)−j∑i=0
MiP(i)n (0)
n−1∑k=n−j−1
c(k,i)Lα+j+1k (x), (25)
and using (20), we have
Lαn(x) =
j+1∑k=0
(j + 1
k
)(n)−kL
α+j+1n−k (x)−
n−1∑k=n−j−1
[j∑i=0
c(k,i)Mi
(Lαn
)(i)
(0)
]Lα+j+1k (x).
As a consequence, we have the following theorem.
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48 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
Theorem 3. Let{Lα+j+1t (x)
}∞t=0
be the sequence Laguerre polynomials withparameter α+j+1. Then, the Laguerre-Sobolev type polynomials can be writtenas
Lαn(x) =
j+1∑k=0
A[k]n L
α+j+1n−k (x), (26)
where, for k = 1, 2, . . . , j + 1, A[k]n are constants of the form
A[k]n = −
j∑i=0
c(n−k;i)Mi
(Lαn
)(i)
(0) +
(j + 1
k
)(n)−k,
and A[0]n = 1.
4. The Recurrence Formula
The Fourier expansion of the polynomial xj+1Lαn(x) is given by
xj+1Lαn(x) = Lαn+j+1(x) +
n+j∑k=0
λn,kLαk (x), (27)
where
λn,k =
⟨xj+1Lαn(x), Lαk (x)
⟩S,α
‖Lαk‖2S,α.
Now, for each p, q ∈ P and each i ∈ {0, . . . , j}, we have
(xj+1p(x)
)(i)(0) =
i∑s=0
(j + 1)!
(j + 1− s)!
(i
s
)[xj+1−sp(i−s)(x)
]x=0
= 0
and thus ⟨xj+1p(x), q(x)
⟩S,α
=⟨p(x), q(x)
⟩α+j+1
.
Furthermore, ⟨p(x), xj+1q(x)
⟩S,α
=⟨xj+1p(x), q(x)
⟩S,α.
It follows from the previous observations that
λn,k =
⟨Lαn(x), xj+1Lαk (x)
⟩S,α
‖Lαk‖2S,α=
⟨Lαn(x), Lαk (x)
⟩α+j+1
‖Lαk‖2S,α.
Therefore, for k ∈ {0, . . . , n − (j + 2)}, we have λn,k = 0. Using the previousequation and (26), we get
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 49
⟨Lαn(x), Lαk (x)
⟩α+j+1
=
j+1∑s=0
A[s]n
⟨Lα+j+1n−s (x), Lαk (x)
⟩α+j+1
=
j+1∑s=0
j+1∑t=0
A[s]n A
[t]k
⟨Lα+j+1n−s (x), Lα+j+1
k−t (x)⟩α+j+1
=
j+1∑s=n−k
A[s]n A
[k+s−n]k
∥∥Lα+j+1n−s
∥∥2
α+j+1
for k ∈ {n − (j + 1), n − j, . . . , n, . . . , n + j − 1, n + j}. As a consequence, wehave the following result.
Theorem 4. For n ∈ N, we have
xj+1Lαn(x) = Lαn+j+1(x) +
n+j∑k=max{0,n−j−1}
λn,kLαk (x),
where
λn,k =
j+1∑s=n−k
A[s]n A
[k+s−n]k
∥∥Lα+j+1n−s
∥∥2
α+j+1∥∥Lαk∥∥2
S,α
.
5. Holonomic Equation
In order to find the second order differential equation satisfied by the Laguerre-Sobolev type orthogonal polynomials, we obtain from (9), for 0 ≤ k ≤ j + 1,
x(Lα+j+1n−k (x)
)′′+ (α+ j + 2− x)
(Lα+j+1n−k (x)
)′= −(n− k)Lα+j+1
n−k (x).
Thus,
xA[k]n
(Lα+j+1n−k (x)
)′′+(α+j+2−x)A[k]
n
(Lα+j+1n−k (x)
)′= −(n−k)A[k]
n Lα+j+1n−k (x)
and, as a consequence, for 0 ≤ k ≤ j + 1,
j+1∑k=0
[xA[k]
n
(Lα+j+1n−k (x)
)′′+ (α+ j + 2− x)A[k]
n
(Lα+j+1n−k (x)
)′]= x
(Lαn(x)
)′′+ (α+ j + 2− x)
(Lαn(x)
)′= −
j+1∑k=0
A[k]n (n− k)Lα+j+1
n−k (x). (28)
Our goal now is to express Lα+j+1n−k (x) as a combination of
(Lαn(x)
)′and
Lαn(x), with rational functions as coefficients. First, we will show that Lα+j+1n−k (x)
can be written as a combination of the Laguerre polynomials Lα+j+1n (x) and
Lα+j+1n−1 (x).
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50 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
Theorem 5. For each k, n ∈ N and k ≤ n, we have
Lαn−k(x) = Dn−k(x)Lαn−1(x) + Sn−k(x)Lαn(x), (29)
where
Dn−k(x) =(x−Mn−k+1)Dn−k+1(x)
Nn−k+1− Dn−k+2(x)
Nn−k+1
and
Sn−k(x) =(x−Mn−k+1)Sn−k+1(x)
Nn−k+1− Sn−k+2(x)
Nn−k+1,
with Dn−1(x) = 1, Dn(x) = 0, Sn−1(x) = 0, Sn(x) = 1, Mn−t =(2(n − t) +
1 + α)
and Nn−t = (n− t)(n− t+ α).
Proof. We will use induction on k. It is clear that the statement holds fork = 1. Assume that the property is valid for all t ≤ k. Using (5), we get
xLαn−k(x) = Lαn−k+1(x) +Mn−kLαn−k(x) +Nn−kL
αn−k−1(x).
Thus,
Lαn−k−1(x) =(x−Mn−k)
Nn−kLαn−k(x)− 1
Nn−kLαn−k+1(x) =
(x−Mn−k)
Nn−k
[Dn−k(x)Lαn−1(x) + Sn−k(x)Lαn(x)
]−
1
Nn−k
[Dn−k+1(x)Lαn−1(x) + Sn−k+1(x)Lαn(x)
]=[(
x−Mn−k(x))
Nn−kDn−k(x)− Dn−k+1(x)
Nn−k
]Lαn−1(x)+[
(x−Mn−k)
Nn−kSn−k(x)− Sn−k+1(x)
Nn−k
]Lαn(x) =
Dn−k−1(x)Lαn−1(x) + Sn−k−1(x)Lαn(x). �X
Now we will prove a property about of the polynomialsDn−k(x) and Sn−k(x)that we will need later.
Theorem 6. For each k, n ∈ N and k ≤ n, Dn−k(x) and Sn−k(x) are polyno-mials with degree k − 1 and k − 2, respectively.
Proof. If k = 1, Dn−1(x) = 1. So deg(1) = 0. Assume that the property isvalid for every t ≤ k. We have that
Dn−(k+1)(x) =(x−Mn−k)Dn−k(x)
Nn−k− Dn−k+1(x)
1Nn−k.
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 51
From the induction hypothesis, deg(Dn−k(x)
)= k−1 and deg
(Dn−k+1(x)
)=
k − 2. Thus
deg(Dn−k−1(x)
)= 1 + deg
(Dn−k(x)
)= k.
The proof for Sn−k(x) is similar. �X
Now, we proceed to write the Laguerre polynomials Lα+j+1n (x) and Lα+j+1
n−1 (x)
as a combination of the polynomials(Lαn(x)
)′and Lαn(x). According to (26) and
(29), we have
Lαn(x) =
j+1∑k=0
A[k]n L
α+j+1n−k (x)
=
j+1∑k=0
A[k]n
[Dn−k(x)Lα+j+1
n−1 (x) + Sn−k,xLα+j+1n (x)
]
=
[j+1∑k=0
A[k]n Dn−k(x)
]Lα+j+1n−1 (x) +
[j+1∑k=0
A[k]n Sn−k(x)
]Lα+j+1n (x). (30)
If we denote
hj(x) =
[j+1∑k=0
A[k]n Dn−k(x)
]and fj−1(x) =
[j+1∑k=0
A[k]n Sn−k(x)
], (31)
then we have
Lαn(x) = fj−1(x)Lα+j+1n (x) + hj(x)Lα+j+1
n−1 (x). (32)
Taking derivatives with respect to x in (30), multiplying by x and applying(10), we get
x(Lαn(x)
)′=
j+1∑k=0
A[k]n x(Lα+j+1n−k (x)
)′=
j+1∑k=0
A[k]n
[(n− k)Lα+j+1
n−k (x) + (n− k)(n− k + α+ j + 1)Lα+j+1n−k−1(x)
]
and, applying (29) on the previous equation, we obtain
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52 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
x(Lαn(x)
)′=
j+1∑k=0
A[k]n
[(n− k)
(Dn−k(x)Lα+j+1
n−1 (x) + Sn−k(x)Lα+j+1n (x)
)+ (n− k)(n− k + α+ j + 1)
(Dn−k−1(x)Lα+j+1
n−1 (x) + Sn−k−1(x)Lα+j+1n (x)
)]=
j+1∑k=0
A[k]n
[(n− k)Dn−k(x)Lα+j+1
n−1 (x)+
(n− k)(n− k + α+ j + 1)Dn−k−1(x)Lα+j+1n−1 (x)
]+
j+1∑k=0
A[k]n
[(n− k)Sn−k(x)Lα+j+1
n (x)+
(n− k)(n− k + α+ j + 1)Sn−k−1(x)Lα+j+1n (x)
].
Thus,
x(Lαn(x)
)′=
j+1∑k=0
[A[k]n (n−k)Dn−k(x) +A[k]
n (n−k)(n−k+α+ j+ 1)Dn−k−1(x)]Lα+j+1n−1 (x)
+
j+1∑k=0
[A[k]n (n−k)Sn−k(x)+A[k]
n (n−k)(n−k+α+j+1)Sn−k−1(x)]Lα+j+1n (x).
If we define
gj+1(x) =
j+1∑k=0
A[k]n
[(n− k)Dn−k(x) + (n− k)(n− k + α+ j + 1)Dn−k−1(x)
]and
qj(x) =
j+1∑k=0
A[k]n
[(n− k)Sn−k(x) + (n− k)(n− k+α+ j+ 1)Sn−k−1(x)
], (33)
then
x(Lα(x)
)′= qj(x)Lα+j+1
n (x) + gj+1(x)Lα+j+1n−1 (x). (34)
From (32) and (34), we obtain the following system of equations:{Lαn(x) = fj−1(x)Lα+j+1
n (x) + hj(x)Lα+j+1n−1 (x);
x(Lα(x)
)′= qj(x)Lα+j+1
n (x) + gj+1(x)Lα+j+1n−1 (x).
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 53
Solving it, we obtain
Lα+j+1n (x) =
Lαn(x)gj+1(x)− x(Lαn(x)
)′hj(x)
fj−1(x)gj+1(x)− hj(x)qj(x), (35)
Lα+j+1n−1 (x) =
x(Lαn(x)
)′fj−1(x)− Lαn(x)qj(x)
fj−1(x)gj+1(x)− hj(x)qj(x)(36)
and, from (28), we get
x(Lαn(x)
)′′+ (α+ j + 2−)
(Lαn(x)
)′=
−j+1∑k=0
A[k]n (n− k)
[Dn−k(x)Lα+j+1
n−1 (x) + Sn−k(x)Lα+j+1n (x)
]=
−
[j+1∑k=0
A[k]n (n−k)Dn−k(x)
]Lα+j+1n−1 (x)−
[j+1∑k=0
A[k]n (n−k)Sn−k(x)
]Lα+j+1n (x).
Applying (35) and (36) on the previous expression, we have
x(Lαn(x)
)′′+ (α+ j + 2− x)
(Lαn(x)
)′=
−
[j+1∑k=0
A[k]n (n− k)Dn−k(x)
](x(Lαn(x)
)′fj−1(x)− Lαn(x)qj(x)
fj−1(x)gj+1(x)− hj(x)qj(x)
)
−
[j+1∑k=0
A[k]n (n− k)Sn−k(x)
](Lαn(x)gj+1(x)− x
(Lαn(x)
)′hj(x)
fj−1(x)gj+1(x)− hj(x)qj(x)
).
Thus,
x(Lαn(x)
)′′+
[(α+ j + 2− x) +
xfj−1(x)∑j+1k=0A
[k]n (n− k)Dn−k(x)
fj−1(x)gj+1(x)− hj(x)qj(x)−
xhj(x)∑j+1k=0A
[k]n (n− k)Sn−k(x)
fj−1(x)gj+1(x)− hj(x)qj(x)
](Lαn(x)
)′+[
−qj(x)
∑j+1k=0A
[k]n (n− k)Dn−k(x)
fj−1(x)gj+1(x)− hj(x)qj(x)+
gj+1(x)∑j+1k=0A
[k]n (n− k)Sn−k(x)
fj−1(x)gj+1(x)− hj(x)qj(x)
]Lαn(x) = 0.
As a consequence, we have proved the following theorem.
Theorem 7. Let{Lαn(x)
}∞n=0
be the Laguerre polynomials and{Lαn(x)
}∞n=0
the orthogonal polynomials associated with the inner product (2).If we consider the same notation as in (29), (31) and (33), then
Revista Colombiana de Matematicas
54 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
Q(x;n)(Lαn(x)
)′′+W (x;n)
(Lαn(x)
)′+R(x;n)Lαn(x) = 0, (37)
where
Q(x;n) = x[fj−1(x)gj+1(x)− hj(x)qj(x)
],
W (x;n) =[(α+ j + 2− x)
[fj−1(x)gj+1(x)− hj(x)qj(x)
]+
xfj−1(x)
j+1∑k=0
A[k]n (n− k)Dn−k(x)− xhj(x)
j+1∑k=0
A[k]n (n− k)Sn−k(x)
]and
R(x;n) =[− qj(x)
j+1∑k=0
A[k]n (n− k)Dn−k(x) + gj+1(x)
j+1∑k=0
A[k]n (n− k)Sn−k(x)
]. (38)
6. An Electrostatic Model
As an application of (37), we present an electrostatic model that responds tothe zeros distribution of the Laguerre-Sobolev type orthogonal polynomials. Forn ∈ N, let {xn,k}1≤k≤n be the zeros of Lαn(x). Thus, for every k ∈ {1, . . . , n},from (37) we get
Q(xn,k, n)(Lαn(xn,k)
)′′+W (xn,k, n)
(Lαn(xn,k)
)′= 0
or, equivalently, (Lαn(xn,k)
)′′(Lαn(xn,k)
)′ = −W (xn,k, n)
Q(xn,k, n). (39)
Therefore,
W (xn,k, n)
Q(xn,k, n)=
(α+ j + 2)
xn,k− 1 +
fj−1(xn,k)∑j+1k=0A
[k]n (n− k)Dn−k(xn,k)
fj−1(xn,k)gj+1(xn,k)− fj(xn,k)gj(xn,k)
−fj(xn,k)
∑j+1k=0A
[k]n (n− k)Sn−k(xn,k)
fj−1(xn,k)gj+1(xn,k)− fj(xn,k)gj(xn,k). (40)
Since F (xn,k) := fj−1(xn,k)gj+1(xn,k) − fj(xn,k)gj(xn,k) is a polynomialof degree at most 2j − 2, then there exists t1 ∈ {1, 2, . . . , 2j − 2}, such that
x(F )1 , . . . , x
(F )t1 are all the zeros of F (x) different from 0.
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 55
In the general case, if F (x) has complex and real zeros, we consider I1 ⊂{0, 1, . . . , t1} such that, for each i ∈ I1, x
(F )i is a complex number (but not real)
and, for each i ∈ J1, x(F )i is a real number, where
J1 = {0, 1, . . . , t1} − I1.
Since for each i ∈ I1, the complex conjugate of x(F )i is also a zero of F (x),
then the number of elements in I1 is even, say 2n, where n is an integer smallerthan or equal to j − 1. If i1, . . . , i2n denotes an order of I1 such that, for
every k ∈ {1, 3, . . . , 2n− 1}, x(F )ik+1
is complex conjugate of x(F )ik
then we define
I1 ={ik ∈ I1 : (1 ≤ k ≤ 2n)∧ (k odd)
}. For each i ∈ I1 and ` ∈ J1, there exist
real constants A, Ai, Bi, Ci, A`, such that(Lαn(xn,k)
)′′(Lαn(xn,k)
)′ =A
xn,k+∑i∈I1
Aix+Bi(xn,k)2 + C2
i
+∑`∈J1
A`
xn,k − x(F )`
= 0
and, since (see [21]),(Lαn(xn,k)
)′′(Lαn(xn,k)
)′ = −2
n∑j=1, j 6=k
1
xn,j − xn,k, (41)
we obtain,
n∑j=1, j 6=k
1
xn,j − xn,k+
1
2
A
xn,k+
1
2
∑i∈I1
Aix+Bi(xn,k)2 + C2
i
+1
2
∑`∈J1
A`
xn,k − x(F )`
= 0. (42)
Therefore, we can make the following electrostatic interpretation of the zerosof Ln(x). If we have n ≥ 1 electric particles located on the complex plane underthe interaction of a logarithmic external field and we have
ϕ1(x) =1
2A ln |x|+ 1
2
∑`∈J1
A` ln∣∣∣x− x(F )
`
∣∣∣+1
2
∑i∈I1
[Ai ln
√x2 + C2
i +BiCi
tan−1
(x
Ci
)],
then (42) indicates that the totally energy gradient
ε(X) = −∑
1≤j<i≤n
ln |xj − xi|+n∑i=1
ϕ1(xi),
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56 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
with X = (x1, . . . , xn), vanishes at the points (xn,1, . . . , xn,n). In other words,
the set of the zeros of Laguerre-Sobolev type orthogonal polynomial Lαn(x)constitutes a critical point. However, it is an open problem to determine if thiscritical point is a maximum, a minimum, or neither.
For a more general study of the electrostatic interpretation of standardorthogonal polynomials, we refer the reader to [6], [7], [8] and [11].
7. Hypergeometric Representation
Our goal in this section is to express the polynomials Lαn(x) in terms of ahypergeometric function, as occurs with the Laguerre polynomials. We know
from (26) that Lαn(x) =∑j+1k=0A
[k]n L
α+j+1n−k (x). So, taking into account (6), we
have
Lαn(x) =
j+1∑k=0
A[k]n (−1)n−k(α+ j + 2)n−k
∞∑t=0
(−n+ k)t(α+ j + 2)t
xt
t!, (43)
or, equivalently,
(−1)n(α+ j + 2)n ×∞∑t=0
(−n)t(α+ j + 2)t
xt
t!×[
j+1∑k=0
A[k]n (−1)k(−n+ t)(−n+ t+ 1) · · · (−n+ k + t− 1)
(α+ j + 2 + n− k) · · · (α+ j + 2 + n− 1)(−n) · · · (−n+ k − 1)
]. (44)
If we define
π(t) =
j+1∑k=0
A[k]n (−1)k(−n+ t)(−n+ t+ 1) · · · (−n+ k + t− 1)
(α+ j + 2 + n− k) · · · (α+ j + 2 + n− 1)(−n) · · · (−n+ k − 1),
which is a polynomial such that deg π(t) = j + 1, then
π(t) = Dn(t+ ξ1)(t+ ξ2) · · · (t+ ξj+1),
where Dn is the leading coefficient of π(t) and ξ1, · · · , ξj+1 are complex num-bers. Therefore,
Lαn(x) =
(−1)n(α+ j + 2)nDn
∞∑t=0
(−n)t(α+ j + 2)t
xt
t!(t+ ξ1)(t+ ξ2) · · · (t+ ξj+1) =
(−1)n(α+ j + 2)nDnξ1 · · · ξj+1
∞∑t=0
(−n)t(1 + ξ1)t · · · (1 + ξj+1)t(α+ j + 2)t(ξ1)t · · · (ξj+1)t
xt
t!. (45)
As a consequence,
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DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 57
Theorem 8. For all n ∈ N,
Lαn(x) = (−1)n(α+ j + 2)nDn(ξ1 · · · ξj+1)
j+2Fj+2(−n, 1 + ξ1, . . . , 1 + ξj+1;α+ j + 2, ξ1, . . . , ξj+1;x).
In other words, the polynomials Lαn(x) can be expressed as a hypergeometricfunction of the form j+2Fj+2.
In [5], the reader can find the hypergeometric representation of the Laguerre-Sobolev type orthogonal polynomials for j = 1, which coincides with the pre-vious formula.
8. Zeros
In order to study the behavior of the zeros of the Laguerre-Sobolev type or-thogonal polynomials, we will need the next proposition. Its proof can be foundin [1].
Proposition 9. Let{Lαn(x)
}∞n=0
be the sequence of monic orthogonal polyno-
mials with respect to the inner product (2). Then, for all n > n, Lαn(x) hasn− n changes of sign in the support of the measure, where n is the number ofmasses in (2).
We see that if Mi > 0 for i = 1, . . . , j then the previous proposition showsthat Lαn(x) has at least (n− j + 1) zeros in the support of the measure. In thefollowing theorem, we show that there exist at least (n−j) zeros in the supportof the measure. First, we present a lemma.
Lemma 10. Let x1, . . . , xk be the different zeros with odd multiplicity of thepolynomial Lαn(x) in (0,∞). Then, there exists ϕ(x) such that degϕ(x) = j,and if ρ(x) = (x− x1) · · · (x− xk) then (ϕρ)(i)(0) = 0 for i = 1, . . . , j.
Proof.
Existence. Let h(x) = ρ(x)(∑j
t=0 atxt)
and such that h(0) = 1. Then, from
the Leibniz rule, we have
h(i)(x) =
i∑k=0
(i
k
)( j∑t=0
atxt
)(k)
ρ(i−k)(x).
Then,
h(i)(0) =
i∑k=0
i!
(i− k)!akρ
(i−k)(0) = 0 for 0 < i ≤ j,
and thus we have
Revista Colombiana de Matematicas
58 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
h(0)(0) = a0ρ(0)(0) = 1
h(1)(0) =1!
1!a0ρ
(1)(0) +1!
0!a1ρ
(0)(0) = 0
h(2)(0) =2!
2!a0ρ
(2)(0) +2!
1!a1ρ
(1)(0) +2!
0!a2ρ
(0)(0) = 0
...
h(j)(0) =j!
j!a0ρ
(j)(0) +j!
(j − 1)!a1ρ
(j−1)(0) + · · ·+
j!
1!aj−1ρ
(1)(0) +j!
0!ajρ
(0)(0) = 0
or, equivalently, a linear system of equations of the form Aa=y,
ρ(0)(0)
ρ(1)(0) ρ(0)(0)
ρ(2)(0) 2ρ(1)(0) 2ρ(0)(0)...
......
ρ(j)(0) j!ρ(j−1)(0)(j−1)! · · · j!ρ(0)(0)
a0
a1
a2
...
aj
=
1
0
0...
0
. (46)
And since detA =∏jt=0 t!ρ
(0)(0) 6= 0, then the polynomial coefficients are
uniquely determined. Thus, we define ϕ(x) =∑jt=0 atx
t.
Degree. This is a particular case of Lemma 2.1 in [1].
Zeros of ϕ(x). Using the same argument, we can show that the zeros of ϕ(x)are outside of (0,∞) ([1]). �X
Theorem 11. If Mi > 0, for i = 0, . . . , j, there exist (n − j) zeros in thesupport of the measure.
Proof. From the previous lemma, we know that ϕ(x) has their zeros outside
of the support of the measure. Then, h(x)Lαn(x) = r(x)ϕ(x)ρ2(x), where r(x)and ϕ(x) do not change sign in (0,∞). Therefore, we have
h(x)Lαn(x) ≥ 0 for x ∈ (0,∞) or h(x)Lαn(x) ≤ 0 for x ∈ (0,∞).
Thus, by the continuity of the polynomial h(x)Lαn(x) and the intermediate-value theorem we have that
h(0)Lαn(0) = 0 or sgn
(∫ ∞0
h(x)Lαn(x) dµ
)= sgn
(h(0)Lαn(0)
).
Volumen 47, Numero 1, Ano 2013
DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 59
As a consequence,∣∣∣∣ ∫ ∞0
h(x)Lαn(x)xe−x dx+ h(0)Lαn(0)
∣∣∣∣ > 0
and, therefore, deg h(x) = j + k ≥ deg Lαn(x) = n. That is, k ≥ n− j, showing
that Lαn(x) has at least n− j zeros in the support of the measure. �X
9. Some Numerical Experiments about Zeros.
Using (26), we consider some examples of Laguerre-Sobolev type orthogonalpolynomials in order to show the behavior of their zeros. First we analyze anexample where for some fixed parameters, the Laguerre-Sobolev type polyno-mials remain unchanged if we change one of the masses.
Example 12. Let M = M0, M1 = 1, n = 3, j = 1 and α = 2. Then, using
MatLab software, we can find that K(0,0)3 (0, 0) = 5, K(1,0)(0, 0) = K
(0,1)3 (0, 0) =
−2.5, K(1,1)(0, 0) = 1.5, L(0)3 (0) = −60 and L
(1)3 (0) = 60.
Then, from (17) we have{(1 + 5M)
(L2
3
)(0)(0)− 2.5
(L2
3
)(1)(0) = −60
−2.5M(L2
3
)(0)(0) + (1 + 1.5)
(L2
3
)(1)(0) = 60
,
and making the computations, the solutions are{(L2
3
)(1)(0) = 24 and
(L2
3
)(0)(0) = 0 if M 6= − 2
5(L2
3
)(1)(0) = − 2
5
(L2
3
)(0)(0) + 24 if M = − 2
5
.
Since M0 ≥ 0, then(L2
3
)(0)(0) = 0 and
(L2
3
)(1)(0) = 24. Furthermore, we
know from (24) and (26) that
c(k;i) =
2∑m=i
2∑t=0
(2
t
)(−1)m+im!
(m− t)!Γ(3 + i)(m− i)!δk,m−t,
and
A[k]n = −
1∑i=0
c(3−k;i)MiP(i)n (0) +
(2
k
)(3)−k.
Therefore,
A[k]3 = −24c(3−k;1) +
(j + 1
k
)(3)−k,
where
Revista Colombiana de Matematicas
60 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
c(3−1;1) = − (1)3
(1)!Γ(4)= −1
6and
c(3−2;1) =
(2
0
)(−1)21!
(1)!Γ(4)(0)!+
(2
1
)(−1)32!
(1)!Γ(4)(1)!= −1
2.
As a consequence,
A[0]3 = 1, A
[1]3 = −24
(− 1
6
)+
(2)(3!)
(2)!= 10 and
A[2]3 = −24
(− 1
2
)+
(2
2
)3!
(1)!= 18
and, from (11),
L43−k(x) =
3−k∑t=0
(−1)3−k+t(3− k)!Γ(3− k + 4 + 1)xt
Γ(3− k − t+ 1)Γ(4 + t+ 1)t!,
that is,
L23(x) = L4
3(x) + 10L42(x) + 18L4
1(x) = x3 − 11x2 + 24x.
In other words, the Laguerre-Sobolev type orthogonal polynomial L23(x)
does not depend on M0.
Example 13. In order to see how the zeros of Lα4 (x) behave when the valuesof the masses vary, we present some computations performed in MatLab. It isimportant to remark that the results of the numerical experiments illustratesthe Proposition 8 and Theorem 3. For n = 4 and several choices of α, wedeveloped some numerical experiments which allow us to conjecture that:
(1) If the mass M0 increases and M1 is fixed, the last zero (arranged indecreasing order) increases.
(2) If the mass M1 increases and M0 is fixed, the last zero (arranged indecreasing order) decreases.
Now, setting j = 1, n = 4, and fixing one of the masses, our goal is todetermine the value of the other mass such that the polynomial Lαn(x) has anegative zero. We will do this for fixed values of α = 3 and α = −0.5.
The following tables show the variation of the last zero of Lα4 (x) when oneof the masses is fixed and the other one varies. The approximated “critical”value of the varying mass (the value that causes a negative zero) is shown onthe caption of the corresponding table.
Volumen 47, Numero 1, Ano 2013
DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 61
M0
Lα 4
(x)
M1
=1
α=
3Z
eros
0.1
x4−
22.7
704x
3+
147.5
038x
2−
255.6
271x−
34.
5442
12.8
667;
6.9666;
3.0629;−
0.1258
0.5
x4−
22.8
096x
3+
148.4
729x
2−
262.6
052x−
20.
2004
12.8
648;
6.9636;
3.0551;−
0.0738
1x
4−
22.8
285x
3+
148.9
393x
2−
265.9
631x−
13.
2982
12.8
638;
6.9621;
3.0512;−
0.0487
10x
4−
22.8
598x
3+
149.7
122x
2−
271.5
277x−
1.8
598
12.8
623;
6.9596;
3.0447;−
0.0068
100
x4−
22.8
643x
3+
149.8
248x
2−
272.3
382x−
0.1
937
12.8
621;
6.9593;
3.0437;−
0.0007
1000
x4−
22.8
648x
3+
149.8
365x
2−
272.4
230x−
0.0
195
12.8
620;
6.9592;
3.0436;−
0.0001
104
x4−
22.8
649x
3+
149.8
377x
2−
272.4
315x
12.
8620;
6.9592;
3.0436;
0
Table
1.M
1=
1;M
0≈
104.
Revista Colombiana de Matematicas
62 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
M1
Lα 4
(x)
M0
=1
α=
3Z
eros
0.1
x4−
24.3
286x
3+
175.7
100x
2−
388.7
389x
+103.0
158
13.2
109;
7.3279;
3.4844;
0.3054
0.5
x4−
23.5
200x
3+
161.2
800x
2−
322.5
600x
+40.3
200
13.0
123;
7.1213;
3.2526;
0.1338
0.85
74x
4−
22.9
997x
3+
151.9
942x
2−
279.9
734x
12.
8989;
6.9999;
3.1009;
0
1x
4−
22.8
285x
3+
148.9
393x
2−
265.9
631x−
13.2
982
12.8
638;
6.9621;
3.0512;−
0.0487
10x
4−
20.3
604x
3+
104.8
934x
2−
63.9
594x−
204.6
701
12.4
594;
6.5268;
2.4159;−
1.0418
100
x4−
19.6
699x
3+
92.5
699x
2−
7.44
13x−
258.2
135
12.3
734;
6.4375;
2.2805;−
1.4215
1000
x4−
19.5
882x
3+
91.1
123x
2−
0.75
65x−
264.5
465
12.3
639;
6.4277;
2.2658;−
1.4692
104
x4−
19.5
799x
3+
90.9
639x
2−
0.07
58x−
265.1
914
12.3
629;
6.4267;
2.2643;−
1.4740
Table
2.M
0=
1;M
0≈
0.8
574.
Volumen 47, Numero 1, Ano 2013
DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 63
M0
Lα 4
(x)
M1
=1
α=−
0.5
Zer
os
0.1
x4−
11.4
932x
3+
28.7
232x
2−
4.51
11x−
4.0701
7.9670;
3.2871;
0.5315;−
0.2
924
0.5
x4−
11.4
652x
3+
28.5
699x
2−
4.76
02x−
3.2078
7.9556;
3.2737;
0.4882;−
0.2
523
1x
4−
11.4
434x
3+
28.4
505x
2−
4.95
42x−
2.5361
7.9467;
3.2633;
0.4504;−
0.2
171
10x
4−
11.3
783x
3+
28.0
941x
2−
5.53
31x−
0.5318
7.9206;
3.2325;
0.2955;−
0.0
703
100
x4−
11.3
630x
3+
28.0
102x
2−
5.66
95x−
0.0597
7.9146;
3.2253;
0.2332;−
0.0
100
1000
x4−
11.3
613x
3+
28.0
006x
2−
5.68
50x−
0.0060
7.9139;
3.2245;
0.2239;−
0.0
011
105
x4−
11.3
611x
3+
27.9
996x
2−
5.68
67x
7.9138;
3.2244;
0.2229;
0
Table
3.M
1=
1;M
0≈
105.
Revista Colombiana de Matematicas
64 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
M1
Lα 4
(x)
M0
=1
α=−
0.5
Zer
os
0.1
x4−
12.5
752x
3+
38.9
473x
2−
24.9
535x
+0.1
805
8.1
928;
3.5196;
0.8555;
0.0073
0.1
122
x4−
12.5
017x
3+
38.2
656x
2−
23.6
548x
8.1746;
3.5004;
0.8264;
0
0.5
x4−
11.6
684x
3+
30.5
375x
2−
8.93
06x−
1.9
960
7.9904;
3.3078;
0.5164;−
0.1
462
1x
4−
11.4
434x
3+
28.4
505x
2−
4.95
42x−
2.5
361
7.9467;
3.2633;
0.4504;−
0.2
171
10x
4−
11.1
941x
3+
26.1
387x
2−
0.54
96x−
3.1
344
7.9010;
3.2176;
0.3909;−
0.3
154
100
x4−
11.1
661x
3+
25.8
794x
2−
0.05
56x−
3.2
015
7.8960;
3.2127;
0.3851;−
0.3
277
1000
x4−
11.1
633x
3+
25.8
532x
2−
0.00
56x−
3.2
083
7.8955;
3.2122;
0.3846;−
0.3
289
104
x4−
11.1
630x
3+
25.8
505x
2−
0.00
06x−
3.2
090
7.8955;
3.2121;
0.3845;−
0.3
291
Table
4.M
0=
1;M
0≈
0.1
122.
Volumen 47, Numero 1, Ano 2013
DIAGONAL GENERAL CASE OF LAGUERRE-SOBOLEV POLYNOMIALS 65
Acknowledgements
The authors thank the valuable comments from the referees. They greatly con-tributed to improve the presentation of the manuscript. The third and fourthauthors were supported by Promep and Conacyt.
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[3] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon andBreach, New York, USA, 1978.
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[5] H. Duenas and F. Marcellan, The Laguerre-Sobolev-Type Orthogonal Poly-nomials. Holonomic Equation and Electrostatic Interpretation, RockyMount. Jour. Math 41 (2011), no. 1, 95–131.
[6] F. Grunbaum, Variations on a Theme of Heine and Stieltjes: An Elec-trostatic Interpretation of the Zeros of Certain Polynomials, Journal ofComp. and App. Math. 99 (1998), 189–194.
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[9] , Classical and Quantum Orthogonal Polynomials in One Variable,Encyclopedia of Mathematics and its Applications, vol. 98, CambridgeUniversity Press, Cambridge, UK, 2005.
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Revista Colombiana de Matematicas
66 HERBERT DUENAS, JUAN C. GARCıA, LUIS E. GARZA & ALEJANDRO RAMıREZ
[12] G. Szego, Orthogonal Polynomials, Colloquium Publications - AmericanMathematical Society, vol. 23, American Mathematical Society, Provi-dence, USA, 4th ed., 1975.
(Recibido en agosto de 2012. Aceptado en enero de 2013)
Departamento de Matematicas
Universidad Nacional de Colombia
Facultad de Ciencias
Carrera 30, calle 45
Bogota, Colombia
e-mail: [email protected]
e-mail: [email protected]
Facultad de Ciencias
Universidad de Colima
Bernal Dıaz del Castillo No. 340
Colonia Villas San Sebastian C.P. 28045
Colima, Colima, Mexico
e-mail: [email protected]
e-mail: [email protected]
Volumen 47, Numero 1, Ano 2013