the diagonal general case of the laguerre-sobolev type

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

Volumen 47, Numero 1, Ano 2013

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:

Revista Colombiana de Matematicas


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,



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)



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



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



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



⟨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.



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



⟨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).



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.



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



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).



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)


).

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)


](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)


]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



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


]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


]. (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.



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),



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,



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



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)

).



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



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.



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.



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.



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.



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.



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.

References

[1] M. Alfaro, G. Lopez, and M. L. Rezola, Some Properties of Zeros ofSobolev-Type Orthogonal Polynomials, Journal of Computational and Ap-plied Mathematics 69 (1996), 171–179.

[2] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia ofMathematics and its Applications, vol. 71, Cambridge University Press,Cambridge, UK, 1999.

[3] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon andBreach, New York, USA, 1978.

[4] H. Duenas and F Marcellan, The Laguerre-SoboleV-Type Orthogonal Poly-nomials, Journal of Approximation Theory 162 (2010), 421–440.

[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.

[7] Mourad E. H. Ismail, An Electrostatic Model for Zeros of General Orthog-onal Polynomials, Pacific J. Math 193 (1999), no. 2, 355–369.

[8] , More on Electrostatics Model for Zeros Of Orthogonal Polyno-mials, Pacific. Journal. Of Numer. Funct. Anal. and Optimiz. 21 (2000),191–204.

[9] , Classical and Quantum Orthogonal Polynomials in One Variable,Encyclopedia of Mathematics and its Applications, vol. 98, CambridgeUniversity Press, Cambridge, UK, 2005.

[10] R. Koekoek, Generalization of the Classical Laguerre Polynomials andsome Q-Analogues, Doctoral dissertation, Techn. Univ. of Delft, Delft,Netherlands, 1990.

[11] F. Marcellan, A. Martınez-Finkelshtein, and P. Martınez-Gonzalez, Elec-trostatic Models for Zeros of Polynomials: Old, New, and some Open Prob-lems, Journal of Comp. and App. Math. 207 (2007), no. 2, 258–272.



[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]


Facultad de Ciencias

Universidad de Colima

Bernal Dıaz del Castillo No. 340

Colonia Villas San Sebastian C.P. 28045

Colima, Colima, Mexico




the diagonal general case of the laguerre-sobolev type

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