The Diagonal General Case of the
Laguerre-Sobolev Type Orthogonal
Polynomials
El caso general diagonal de los polinomios ortogonales de tipo Laguerre-Sobolev
Herbert Due˜
nas
1,B,
Juan C. Garc´ıa
1,
Luis E. Garza
2,
Alejandro Ram´ırez
21
Universidad Nacional de Colombia, Bogot´
a, Colombia
2Universidad de Colima, Colima, M´
exico
Abstract.We consider the family of polynomials orthogonal with respect to the Sobolev type inner product corresponding to the diagonal general case of the Laguerre-Sobolev type orthogonal polynomials. We analyze some proper-ties of these polynomials, such as the holonomic equation that they satisfy and, as an application, an electrostatic interpretation of their zeros. We also obtain a representation of such polynomials as a hypergeometric function, and study the behavior of their zeros.
Key words and phrases. Orthogonal polynomials, Laguerre-Sobolev type poly-nomials, Laguerre Polypoly-nomials, Derivative of a Dirac Delta.
2010 Mathematics Subject Classification.33C47, 42C05.
Resumen. Se considera la familia de los polinomios ortogonales con respecto a un producto interno de tipo Sobolev correspondiente al caso general di-agonal de los polinomios ortogonales de tipo Laguerre-Sobolev. Se analizan algunas propiedades de estos polinomios tales como la ecuaci´on holon´omica que satisfacen y, como una aplicaci´on de dicha ecuaci´on, una interpretaci´on electrost´atica de sus ceros. Tambi´en se obtiene una representaci´on de tales polinomios en t´erminos de una funci´on hipergeom´etrica, 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.
40 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ
1. Introduction
The Laguerre polynomials are defined as the orthogonal polynomials with re-spect to the inner product
hp, qiα=
Z
I
p(x)q(x)dσ(x), (1)
where dσ(x) = xαe−xdx, I is the positive real line, α > −1, and p, q are
polynomials with real coefficients. We denote by
Lα n(x)
∞
n=0 the sequence of monic Laguerre polynomials,
and by e
Lα n(x)
∞
n=0 the sequence of orthogonal polynomials with respect to
the inner product
hp, qiS,α= Z I p(x)q(x)dσ+ j X i=0 Mip(i)(0)q(i)(0), (2)
whereMi ∈R+ andj∈N. The goal of this paper is to generalize some results obtained in [4], [5] and [10], extending them for an arbitrary finite number of masses.
The structure of the manuscript is as follows. In Section 2, we present some preliminary results about the Laguerre polynomials. In Section 3, we give a connection formula which expresses the Laguerre-Sobolev type polynomials in terms 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 differential equation that the Laguerre-Sobolev type polynomials satisfy. This holonomic equation will be used in Section 6 in order to find an electrostatic interpretation of the zeros ofLeαn(x).
In section 7, we use the hypergeometric representation of the Laguerre poly-nomials to construct a hypergeometric representation for the Laguerre-Sobolev type polynomials and finally, in Section 8, we analyze the location of the zeros ofLeαn(x). Some numerical experiments usingMatLabandMathematicasoftware
are presented.
2. Preliminaries
Letµbe a linear moment functional defined inP, the linear space of polynomials with real coefficients, and define their corresponding moments byµn=hµ, xni,
n≥0. If{Pn(x)}n≥0 is a polynomials sequence such that the degree ofPn(x)
isnand it satisfies
µ, Pn(x)Pm(x)
=Knδm,n,
δm,n=
(
0, if m6=n; 1, if n=m,
then we say that
Pn(x) n≥0 is the sequence of orthogonal polynomials
asso-ciated with the linear functionalµ. We will assume thatµis a positive definite linear 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 the orthogonality 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∈Nare real withγn6= 0. Then-th reproducing KernelKn(x, y)
associated with Pn(x) n≥0 is defined by Kn(x, y) = n X k=0 Pk(x)Pk(y) kPkk2µ , wherekPkk2µ= D µ, P(x)2E .
There is an explicit expression forKn(x, y), the so-called Christoffel-Darboux
formula, given by
Kn(x, y) =
Pn+1(x)Pn(y)−Pn(x)Pn+1(y)
(x−y)kPnk2µ
,
for x 6= y. We will use the following notation for the partial derivatives of
Kn(x, y) ∂i+t Kn(x, y) ∂xi∂yt =K (i,t) n (x, y).
Ifp∈Pand deg(p)≤nthen we have
Kn(0,i)(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 to deduce the following formula:
42 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ Kn(0−,j1)(x, a) = j! kPn−1k2(x−a)j+1 × Pn(a) +Pn(1)(a)(x−a) + Pn(2)(a) 2! (x−a) 2+· · ·+P (n) n (a) n! (x−a) n × Pn−1(a) +P (1) n−1(a)(x−a) + Pn(2)−1(a) 2! (x−a) 2+· · ·+P (j) n−1(a) j! (x−a) j − j! kPn−1k2(x−a)j+1 × Pn−1(a) +P (1) n−1(a)(x−a) + Pn(2)−1(a) 2! (x−a) 2+· · ·+P (n−1) n−1 (a) (n−1)! (x−a) n−1 × Pn(a) +Pn(1)(a)(x−a) + Pn(2)(a) 2! (x−a) 2+· · ·+P (j) n (a) j! (x−a) j . (4)
In the following section, we will deduce a formula forKn(i,j−1)(a, a), which will help us to deduce the derivative of the polynomialLeαn(x).
The families of orthogonal polynomials that have been most extensively studied in the literature are the Jacobi, Laguerre and Hermite polynomials. They are the so-called classical orthogonal polynomials, and have (among others) the following characterization:
Pn(x) n∈Nis classical if and only ifPn,n∈N, is a solution of the second order differential equation
φ(x)y00(x) +τ(x)y0(x) +λny(x) = 0,
where φ and τ are polynomials independent ofn, with degree at most 2 and exactly 1, respectively, andλn is a constant.
In this paper, we will focus our attention on theLaguerremonic orthogonal polynomials. In the following proposition we list some of their properties, that will be useful in the upcoming sections (see [2], [3] and [12]).
Proposition 1. Let {Lα
n}n≥0 be the sequence of Laguerre monic orthogonal polynomials. Then,
1) Forn∈N,
xLαn(x) =Lαn+1(x) + (2n+α+ 1)Lnα(x) +n(n+α)Lαn−1(x). (5)
Lαn(x) = (−1)n(α+ 1)n ∞ X 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| Q k=0 (a+ksgn(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) Forn∈N,
(Lαn)(i)(0) = (−1)n+i n!Γ(α+n+ 1)
(n−i)!Γ(α+i+ 1). (7)
4) Forn∈N,
kLαnk2α=n!Γ(n+α+ 1). (8)
5) Forn∈N,Lαn(x)satisfies the differential equation
xy00+ (α+ 1−x)y0 =−ny. (9) 6) Forn∈N, x Lαn(x) 0 =nLαn(x) +n(n+α)Lαn−1(x). (10) 7) Forn∈N, Lαn(x) = n! (−1)n n X k=0 Γ(n+α+ 1) (n−k)!Γ(α+k+ 1) (−x)k k! . (11) 8) Forn∈N, Lαn(x) =Lαn+1(x) +nL α+1 n−1(x). (12) 9) Forn∈N, Lαn0 (x) =nLαn+1−1(x). (13)
44 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ
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 e
Pn(x) n≥0 the sequence of
polynomials orthogonal with respect to the inner product hp, qiS=hp, qiµ+
j
X
i=0
Mip(i)(0)q(i)(0).
Since{Pn}n≥0 is a basis in the linear space of polynomials, we can write
e Pn(x) = n X k=0 an,kPk(x) =Pn(x) + n−1 X k=0 an,kPk(x), (14) where an,k = e Pn(x), Pk(x) µ kPk(x)k2µ .
Taking into account that fork < n
e Pn(x), Pk(x)µ=− j X i=0 MiPen(i)(a)P (i) k (a), e Pn(x) can be written as e Pn(x) =Pn(x)− j X i=0 MiPen(i)(a) n−1 X k=0 Pk(i)(a)Pk(x) kPk(x)k2µ (15) =Pn(x)− j X i=0 MiPen(i)(a)K (0,i) n−1(x, a). (16) In order to findPe (i)
n (a) fori= 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) e Pn(0)(a) e Pn(1)(a) e Pn(2)(a) .. . e Pn(j)(a) = Pn(0)(a) Pn(1)(a) Pn(2)(a) .. . Pn(j)(a) (17)
which is obtained by taking derivatives in (16), where K(p,q) := K(p,q)
n−1(a, a).
Kn(0−,j1)(x, a) = j! kPn−1k2 × " n X r=j+1 Pn(r)(a) r! Pn−1(a)(x−a) r−j−1+· · ·+ Pn(k−)1(a) k! (x−a) k+r−j−1+· · ·+P (j) n−1(a) j! (x−a) r−1 ! − n−1 X k=j+1 Pn(k−)1(a) k! Pn(a)(x−a) k−j−1+· · ·+ Pn(r)(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 Kn(i,j−1)(a, a), we calculate thei-th derivative in (18) with respect toxand evaluate it atx=ato obtain
Kn(i−;j1)(a, a) = j! kPn−1k2 " n X r=j+1 i!P (r) n (a)P (i+j+1−r) n−1 (a) r!(i+j+ 1−r)! − n−1 X 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 polynomialsLeαn(x), orthogonal with respect to the
inner product (2), in terms of some Laguerre orthogonal polynomials. First, we need the following theorem.
Theorem 2. For t, n∈N and α >−1, the Laguerre polynomials satisfy the
following property ([9]) Lαn(x) = t X k=0 t k (n)−kLαn−+tk(x). (20) Let Lα+j+1
n (x) be the sequence of Laguerre polynomials orthogonal with
respect todσ(x) =e−xxα+j+1dx. We will expressKn(0−,i1)(x,0) as a linear com-bination ofLαn+j+1(x). Then, kLα n−1k2α (Lα n−1)(i)(0) Kn(0−,i1)(x,0) =Lαn−+j1+1(x) + n−2 X k=0 bn−1,kL α+j+1 k (x), (21) where
46 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ bn−1,k= kLα n−1k 2 α (Lα n−1)(i)(0) Kn(0−,i1)(x,0), Lαk+j+1(x) α+j+1 kLαk+j+1(x)k2 α+j+1 = kL α n−1k2α (Lα n−1)(i)(0)kL α+j+1 k (x)k 2 α+j+1 D Kn(0−,i1)(x,0), Lαk+j+1(x)E α+j+1.
From (1) and (3), we see that for 0≤k≤n−2−j,
D Kn(0−,i1)(x,0), Lαk+j+1(x)E α+j+1 = Z ∞ 0 Kn(0−,i1)(x,0)Lkα+j+1(x)xα+j+1e−xdx = Z ∞ 0 Kn(0−,i1)(x,0)(xj+1Lkα+j+1(x))xαe−xdx =DKn(0−,i1)(x,0), xj+1Lαk+j+1(x)E α =xj+1Lαk+j+1(x) (i) (0) = 0. In other words, Kn(0−,i1)(x,0) = (L α n−1)(i)(0) kLα n−1(x)k2α Lαn+−j1+1(x)+ n−2 X k=n−j−1 D Kn(0−,i1)(x,0)Lαk+j+1(x)E α+j+1 kLαk+j+1(x)k2 α+j+1 Lαk+j+1(x). (22)
Taking into account that
Kn(0−,i1)(x,0) = n−1 X m=0 Lαm(x) (Lαm)(i)(0) kLα m(x)k2α , (23)
D Kn(0−,i1)(x,0), Lαk+j+1(x)E α+j+1 = n−1 X m=0 (Lα m)(i)(0) kLα m(x)k2α D Lαm(x), Lαk+j+1(x)E α+j+1 = n−1 X m=0 (Lαm)(i)(0) kLα m(x)k2α j+1 X t=0 j+ 1 t (m)(−t) D Lαm+−jt+1(x), Lαk+j+1(x)E α+j+1 = n−1 X m=0 j+1 X t=0 j+ 1 t (m) (−t)(Lαm)(i)(0) kLα m(x)k2α D Lαm+−jt+1(x), L α+j+1 k (x) E α+j+1 = n−1 X m=i j+1 X t=0 j+ 1 t (−1)m+im! (m−t)!Γ(α+i+ 1)(m−i)!kL α+j+1 k (x)k 2 α+j+1δk,m−t. Forn−j−1≤k≤n−2, let c(k;i)= D Kn(0−,i1)(x,0), Lαk+j+1(x)E α+j+1 kLαk+j+1(x)k2 α+j+1 and c(n−1;i)= (Ln−1,α)(i)(0) kLα n−1(x)k2α , or, equivalently, c(k;i)= n−1 X m=i j+1 X 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, e Pn(x) =Lαn(x)− j X i=0 MiPen(i)(0) n−1 X k=n−j−1 c(k,i)L α+j+1 k (x), (25)
and using (20), we have
e Lαn(x) = j+1 X k=0 j+ 1 k (n)−kL α+j+1 n−k (x)− n−1 X k=n−j−1 " j X i=0 c(k,i)Mi e Lαn (i) (0) # Lαk+j+1(x).
48 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ
Theorem 3. Let
Lαt+j+1(x) ∞
t=0 be the sequence Laguerre polynomials with parameterα+j+1. Then, the Laguerre-Sobolev type polynomials can be written as e Lαn(x) = j+1 X k=0 An[k]Lαn−+kj+1(x), (26)
where, for k= 1,2, . . . , j+ 1,A[nk] are constants of the form
A[nk]=− j X i=0 c(n−k;i)Mi e Lαn (i) (0) + j+ 1 k (n)−k, andA[0]n = 1.
4. The Recurrence Formula
The Fourier expansion of the polynomialxj+1Leαn(x) is given by
xj+1Le α n(x) =Le α n+j+1(x) + n+j X k=0 λn,kLe α k(x), (27) where λn,k= xj+1 e Lα n(x),Leαk(x) S,α kLeαkk2S,α .
Now, for eachp, q∈Pand eachi∈ {0, . . . , j}, we have
xj+1p(x)(i) (0) = i X s=0 (j+ 1)! (j+ 1−s)! i s xj+1−sp(i−s)(x) x=0= 0 and thus D xj+1p(x), q(x)E S,α = p(x), q(x) α+j+1. Furthermore, D p(x), xj+1q(x)E S,α =Dxj+1p(x), q(x)E S,α .
It follows from the previous observations that
λn,k = D e Lα n(x), xj+1Leαk(x) E S,α kLeαkk2S,α = D e Lα n(x),Leαk(x) E α+j+1 kLeαkk2S,α .
Therefore, for k∈ {0, . . . , n−(j+ 2)}, we have λn,k = 0. Using the previous
D e Lαn(x),Leαk(x) E α+j+1 = j+1 X s=0 An[s]DLnα−+js+1(x),Leαk(x) E α+j+1 = j+1 X s=0 j+1 X t=0 An[s]A[kt]DLnα−+sj+1(x), Lkα−+tj+1(x)E α+j+1 = j+1 X s=n−k An[s]Ak[k+s−n]L α+j+1 n−s 2 α+j+1 fork ∈ {n−(j+ 1), n−j, . . . , n, . . . , n+j−1, n+j}. As a consequence, we have the following result.
Theorem 4. Forn∈N, we have xj+1Leαn(x) =Leαn+j+1(x) + n+j X k=max{0,n−j−1} λn,kLeαk(x), where λn,k= j+1 X s=n−k A[ns]A[kk+s−n] L α+j+1 n−s 2 α+j+1 eLαk 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,
xLαn+−kj+1(x) 00 + (α+j+ 2−x)Lnα−+jk+1(x) 0 =−(n−k)Lnα−+kj+1(x). Thus, xA[nk] Lαn−+jk+1(x) 00 +(α+j+2−x)A[nk] Lαn+−kj+1(x) 0 =−(n−k)A[nk]L α+j+1 n−k (x)
and, as a consequence, for 0≤k≤j+ 1,
j+1 X k=0 xA[nk] Lαn+−jk+1(x) 00 + (α+j+ 2−x)A[nk] Lαn−+jk+1(x) 0 =xLe α n(x) 00 + (α+j+ 2−x)Leαn(x) 0 =− j+1 X k=0 An[k](n−k)Lnα−+jk+1(x). (28)
Our goal now is to express Lαn+−kj+1(x) as a combination of Leαn(x) 0
and
e
Lαn(x), with rational functions as coefficients. First, we will show thatL α+j+1
n−k (x)
can be written as a combination of the Laguerre polynomials Lα+j+1
n (x) and
50 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ
Theorem 5. For eachk, n∈N andk≤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 andNn−t= (n−t)(n−t+α).
Proof. We will use induction on k. It is clear that the statement holds for
k= 1. Assume that the property is valid for allt≤k. Using (5), we get
xLαn−k(x) =Lαn−k+1(x) +Mn−kLαn−k(x) +Nn−kLαn−k−1(x). Thus, Lαn−k−1(x) =(x−Mn−k) Nn−k Lαn−k(x)− 1 Nn−k Lαn−k+1(x) = (x−Mn−k) Nn−k h Dn−k(x)Lαn−1(x) +Sn−k(x)Lαn(x) i − 1 Nn−k h Dn−k+1(x)Lαn−1(x) +Sn−k+1(x)Lαn(x) i = " x−Mn−k(x) Nn−k Dn−k(x)− Dn−k+1(x) Nn−k # Lαn−1(x)+ (x −Mn−k) Nn−k Sn−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) andSn−k(x)
that we will need later.
Theorem 6. For eachk, n∈Nandk≤n,Dn−k(x) andSn−k(x)are
polyno-mials with degree k−1 andk−2, respectively.
Proof. If k = 1, Dn−1(x) = 1. So deg(1) = 0. Assume that the property is
valid for everyt≤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 forSn−k(x) is similar. X
Now, we proceed to write the Laguerre polynomialsLαn+j+1(x) andL α+j+1
n−1 (x)
as a combination of the polynomials Leαn(x) 0
andLeαn(x). According to (26) and
(29), we have e Lαn(x) = j+1 X k=0 A[nk]Lαn+−jk+1(x) = j+1 X k=0 A[nk] h Dn−k(x)L α+j+1 n−1 (x) +Sn−k,xLαn+j+1(x) i = "j+1 X k=0 A[nk]Dn−k(x) # Lαn−+j1+1(x) + "j+1 X k=0 A[nk]Sn−k(x) # Lαn+j+1(x). (30) If we denote hj(x) = "j+1 X k=0 A[nk]Dn−k(x) # and fj−1(x) = "j+1 X k=0 A[nk]Sn−k(x) # , (31) then we have e Lαn(x) =fj−1(x)Lαn+j+1(x) +hj(x)Lαn−+1j+1(x). (32)
Taking derivatives with respect toxin (30), multiplying byxand applying (10), we get x Leαn(x) 0 = j+1 X k=0 An[k]x Lαn+−kj+1(x)0 = j+1 X k=0 An[k]h(n−k)Lnα+−kj+1(x) + (n−k)(n−k+α+j+ 1)Lnα−+jk+1−1(x)i
52 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ x Leαn(x) 0 = j+1 X k=0 A[nk]h(n−k)Dn−k(x)Lαn+−1j+1(x) +Sn−k(x)Lαn+j+1(x) + (n−k)(n−k+α+j+ 1)Dn−k−1(x)Lαn+−1j+1(x) +Sn−k−1(x)Lαn+j+1(x) i = j+1 X k=0 A[nk]h(n−k)Dn−k(x)L α+j+1 n−1 (x)+ (n−k)(n−k+α+j+ 1)Dn−k−1(x)Lαn−+j1+1(x) i + j+1 X k=0 A[nk]h(n−k)Sn−k(x)Lαn+j+1(x)+ (n−k)(n−k+α+j+ 1)Sn−k−1(x)Lαn+j+1(x) i . Thus, x Leαn(x) 0 = j+1 X k=0 h A[nk](n−k)Dn−k(x) +A[nk](n−k)(n−k+α+j+ 1)Dn−k−1(x) i Lαn+−1j+1(x) + j+1 X k=0 h A[nk](n−k)Sn−k(x)+A[nk](n−k)(n−k+α+j+1)Sn−k−1(x) i Lαn+j+1(x). If we define gj+1(x) = j+1 X k=0 A[nk]h(n−k)Dn−k(x) + (n−k)(n−k+α+j+ 1)Dn−k−1(x) i and qj(x) = j+1 X k=0 A[nk]h(n−k)Sn−k(x) + (n−k)(n−k+α+j+ 1)Sn−k−1(x) i , (33) then x Leα(x) 0 =qj(x)Lnα+j+1(x) +gj+1(x)Lαn−+j1+1(x). (34)
From (32) and (34), we obtain the following system of equations:
( e Lαn(x) =fj−1(x)Lnα+j+1(x) +hj(x)Lαn+−j1+1(x); x Leα(x) 0 =qj(x)Lnα+j+1(x) +gj+1(x)L α+j+1 n−1 (x).
Solving it, we obtain Lαn+j+1(x) =Le α n(x)gj+1(x)−x Leαn(x) 0 hj(x) fj−1(x)gj+1(x)−hj(x)qj(x) , (35) Lαn−+j1+1(x) =x Le α n(x) 0 fj−1(x)−Leαn(x)qj(x) fj−1(x)gj+1(x)−hj(x)qj(x) (36) and, from (28), we get
x Leαn(x) 00 + (α+j+ 2−) Leαn(x) 0 = − j+1 X k=0 A[nk](n−k)hDn−k(x)Lαn+−1j+1(x) +Sn−k(x)Lαn+j+1(x) i = − "j+1 X k=0 A[nk](n−k)Dn−k(x) # Lαn+−1j+1(x)− "j+1 X k=0 A[nk](n−k)Sn−k(x) # Lαn+j+1(x).
Applying (35) and (36) on the previous expression, we have
x Leαn(x) 00 + (α+j+ 2−x) Leαn(x) 0 = − "j+1 X k=0 A[nk](n−k)Dn−k(x) # x Leαn(x) 0 fj−1(x)−Leαn(x)qj(x) fj−1(x)gj+1(x)−hj(x)qj(x) ! − "j+1 X k=0 A[nk](n−k)Sn−k(x) # e Lαn(x)gj+1(x)−x Leαn(x) 0 hj(x) fj−1(x)gj+1(x)−hj(x)qj(x) ! . Thus, x Leαn(x) 00 + (α+j+ 2−x) +xfj−1(x) Pj+1 k=0A [k] n (n−k)Dn−k(x) fj−1(x)gj+1(x)−hj(x)qj(x) − xhj(x)P j+1 k=0A [k] n (n−k)Sn−k(x) fj−1(x)gj+1(x)−hj(x)qj(x) e Lαn(x)0+ −qj(x) Pj+1 k=0A [k] n (n−k)Dn−k(x) fj−1(x)gj+1(x)−hj(x)qj(x) + gj+1(x)P j+1 k=0A [k] n (n−k)Sn−k(x) fj−1(x)gj+1(x)−hj(x)qj(x) e Lαn(x) = 0.
As a consequence, we have proved the following theorem.
Theorem 7. Let
Lα n(x)
∞
n=0 be the Laguerre polynomials and
e
Lα n(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
54 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ Q(x;n) Leαn(x) 00 +W(x;n) Leαn(x) 0 +R(x;n)Leαn(x) = 0, (37) where Q(x;n) =x fj−1(x)gj+1(x)−hj(x)qj(x) , W(x;n) =h(α+j+ 2−x) fj−1(x)gj+1(x)−hj(x)qj(x)+ xfj−1(x) j+1 X k=0 A[nk](n−k)Dn−k(x)−xhj(x) j+1 X k=0 A[nk](n−k)Sn−k(x) i and R(x;n) = " −qj(x) j+1 X k=0 A[nk](n−k)Dn−k(x) +gj+1(x) j+1 X k=0 A[nk](n−k)Sn−k(x) # . (38) 6. An Electrostatic Model
As an application of (37), we present an electrostatic model that responds to the zeros distribution of the Laguerre-Sobolev type orthogonal polynomials. For
n∈N, let {xn,k}1≤k≤n be the zeros of Leαn(x). Thus, for everyk∈ {1, . . . , n},
from (37) we get Q(xn,k, n) Leαn(xn,k) 00 +W(xn,k, n) Leαn(xn,k) 0 = 0 or, equivalently, e Lα n(xn,k) 00 e Lα n(xn,k) 0 =− 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) Pj+1 k=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) Pj+1 k=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 polynomial
of degree at most 2j −2, then there exists t1 ∈ {1,2, . . . ,2j−2}, such that x(1F), . . . , x(tF1) are all the zeros ofF(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 eachi∈I1,x (F)
i is a complex number (but not real)
and, for eachi∈J1,x (F)
i is a real number, where
J1={0,1, . . . , t1} −I1.
Since for eachi∈I1, the complex conjugate ofx (F)
i is also a zero ofF(x),
then the number of elements inI1is even, say 2nb, wherenbis an integer smaller
than or equal to j −1. If i1, . . . , i2bn denotes an order of I1 such that, for
everyk∈ {1,3, . . . ,2nb−1},x(iF)
k+1 is complex conjugate ofx (F)
ik then we define
¨
I1=ik ∈I1: (1≤k≤2nb)∧(kodd) . For eachi∈I¨1and`∈J1, there exist
real constantsA, Ai,Bi,Ci, ˙A`, such that
e Lαn(xn,k) 00 e Lα n(xn,k) 0 = A xn,k +X i∈I¨1 Aix+Bi (xn,k)2+Ci2 +X `∈J1 ˙ A` xn,k−x (F) ` = 0
and, since (see [21]),
e Lα n(xn,k) 00 e Lα n(xn,k) 0 =−2 n X j=1, j6=k 1 xn,j−xn,k , (41) we obtain, n X j=1, j6=k 1 xn,j−xn,k +1 2 A xn,k + 1 2 X i∈I¨ 1 Aix+Bi (xn,k)2+Ci2 +1 2 X `∈J1 ˙ A` xn,k−x (F) ` = 0. (42)
Therefore, we can make the following electrostatic interpretation of the zeros ofLen(x). If we haven≥1 electric particles located on the complex plane under
the interaction of a logarithmic external field and we have
ϕ1(x) = 1 2Aln|x|+ 1 2 X `∈J1 ˙ A`ln x−x (F) ` + 1 2 X i∈I¨ 1 Ailn q x2+C2 i + Bi Ci tan−1 x Ci ,
then (42) indicates that the totally energy gradient
ε(X) =− X 1≤j<i≤n ln|xj−xi|+ n X i=1 ϕ1(xi),
56 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ
withX = (x1, . . . , xn), vanishes at the points (xn,1, . . . , xn,n). In other words,
the set of the zeros of Laguerre-Sobolev type orthogonal polynomial Leαn(x)
constitutes a critical point. However, it is an open problem to determine if this critical point is a maximum, a minimum, or neither.
For a more general study of the electrostatic interpretation of standard orthogonal polynomials, we refer the reader to [6], [7], [8] and [11].
7. Hypergeometric Representation
Our goal in this section is to express the polynomials Leαn(x) in terms of a
hypergeometric function, as occurs with the Laguerre polynomials. We know from (26) that Leαn(x) =
Pj+1
k=0A [k]
n Lαn+−kj+1(x). So, taking into account (6), we
have e Lαn(x) = j+1 X k=0 A[nk](−1)n−k(α+j+ 2)n−k ∞ X t=0 (−n+k)t (α+j+ 2)t xt t!, (43) or, equivalently, (−1)n(α+j+ 2)n× ∞ X t=0 (−n)t (α+j+ 2)t xt t!× "j+1 X k=0 A[nk](−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 X k=0 A[nk](−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 degπ(t) =j+ 1, then
π(t) =Dn(t+ξ1)(t+ξ2)· · ·(t+ξj+1),
where Dn is the leading coefficient of π(t) andξ1,· · · , ξj+1 are complex
num-bers. Therefore, e Lαn(x) = (−1)n(α+j+ 2)nDn ∞ X 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 ∞ X 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 alln∈N,
e
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 Leαn(x) can be expressed as a hypergeometric
function of the formj+2Fj+2.
In [5], the reader can find the hypergeometric representation of the Laguerre-Sobolev type orthogonal polynomials forj = 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 found in [1]. Proposition 9. Let e Lα n(x) ∞
n=0be the sequence of monic orthogonal polyno-mials with respect to the inner product (2). Then, for all n > n, Leαn(x) has
n−n changes of sign in the support of the measure, wheren is the number of masses in (2).
We see that ifMi>0 fori= 1, . . . , j then the previous proposition shows
thatLeαn(x) has at least (n−j+ 1) zeros in the support of the measure. In the
following theorem, we show that there exist at least (n−j) zeros in the support of the measure. First, we present a lemma.
Lemma 10. Let x1, . . . , xk be the different zeros with odd multiplicity of the
polynomial Leαn(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 fori= 1, . . . , j.
Proof.
Existence.Leth(x) =ρ(x)Pj t=0atx
tand such that h(0) = 1. Then, from
the Leibniz rule, we have
h(i)(x) = i X k=0 i k j X t=0 atxt !(k) ρ(i−k)(x). Then, h(i)(0) = i X k=0 i! (i−k)!akρ (i−k)(0) = 0 for 0< i ≤j,
58 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ 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 formAa=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 = Qj
t=0t!ρ
(0)(0) 6= 0, then the polynomial coefficients are
uniquely determined. Thus, we defineϕ(x) =Pj
t=0atxt.
Degree.This is a particular case of Lemma 2.1 in [1].
Zeros ofϕ(x).Using the same argument, we can show that the zeros ofϕ(x)
are outside of (0,∞) ([1]). X
Theorem 11. If Mi > 0, for i = 0, . . . , j, there exist (n−j) zeros in the
support 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)Leαn(x) =r(x)ϕ(x)ρ2(x), wherer(x)
andϕ(x) do not change sign in (0,∞). Therefore, we have
h(x)Leαn(x)≥0 for x∈(0,∞) or h(x)Leαn(x)≤0 for x∈(0,∞).
Thus, by the continuity of the polynomialh(x)Leαn(x) and the
intermediate-value theorem we have that
h(0)Leαn(0) = 0 or sgn Z ∞ 0 h(x)Leαn(x)dµ = sgnh(0)Leαn(0) .
As a consequence, Z ∞ 0 h(x)Leαn(x)xe−xdx+h(0)Leαn(0) >0
and, therefore, degh(x) =j+k≥degLeαn(x) =n. That is,k≥n−j, showing
thatLeαn(x) has at leastn−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 orthogonal polynomials in order to show the behavior of their zeros. First we analyze an example where for some fixed parameters, the Laguerre-Sobolev type polyno-mials remain unchanged if we change one of the masses.
Example 12. LetM =M0, M1 = 1, n= 3, j = 1 and α = 2. Then, using MatLabsoftware, we can find thatK3(0,0)(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 andL (1)
3 (0) = 60.
Then, from (17) we have
( (1 + 5M) Le23 (0) (0)−2.5 Le23 (1) (0) =−60 −2.5M Le23 (0) (0) + (1 + 1.5) Le23 (1) (0) = 60 , and making the computations, the solutions are
( e L2 3 (1) (0) = 24 and Le23 (0) (0) = 0 if M 6=−2 5 e L2 3 (1) (0) =−2 5 Le 2 3 (0) (0) + 24 if M =−2 5 . Since M0 ≥0, then Le23 (0) (0) = 0 and Le23 (1) (0) = 24. Furthermore, we know from (24) and (26) that
c(k;i)= 2 X m=i 2 X t=0 2 t ( −1)m+im! (m−t)!Γ(3 +i)(m−i)!δk,m−t, and A[nk]=− 1 X i=0 c(3−k;i)MiPen(i)(0) + 2 k (3)−k. Therefore, A[3k]=−24c(3−k;1)+ j+ 1 k (3)−k, where
60 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ c(3−1;1)=− (1)3 (1)!Γ(4)=− 1 6 and 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 X 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, e L23(x) =L43(x) + 10L42(x) + 18L41(x) =x3−11x2+ 24x.
In other words, the Laguerre-Sobolev type orthogonal polynomial Le23(x)
does not depend onM0.
Example 13. In order to see how the zeros ofLeα4(x) behave when the values
of the masses vary, we present some computations performed inMatLab. It is important to remark that the results of the numerical experiments illustrates the Proposition 8 and Theorem 3. For n = 4 and several choices of α, we developed some numerical experiments which allow us to conjecture that:
(1) If the mass M0 increases and M1 is fixed, the last zero (arranged in
decreasing order) increases.
(2) If the mass M1 increases and M0 is fixed, the last zero (arranged in
decreasing order) decreases.
Now, setting j = 1, n = 4, and fixing one of the masses, our goal is to determine the value of the other mass such that the polynomial Leαn(x) has a
negative zero. We will do this for fixed values ofα= 3 andα=−0.5.
The following tables show the variation of the last zero of Leα4(x) when one
of 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 on the caption of the corresponding table.
M 0 eL α(4 x ) M 1 = 1 α = 3 Zeros 0 . 1 x 4 − 22 . 7704 x 3 + 147 . 5038 x 2 − 255 . 6271 x − 34 . 5442 12 . 8667; 6 . 9666; 3 . 0629; − 0 . 1258 0 . 5 x 4 − 22 . 8096 x 3 + 148 . 4729 x 2 − 262 . 6052 x − 20 . 2004 12 . 8648; 6 . 9636; 3 . 0551; − 0 . 0738 1 x 4− 22 . 8285 x 3+ 148 . 9393 x 2− 265 . 9631 x − 13 . 2982 12 . 8638; 6 . 9621; 3 . 0512; − 0 . 0487 10 x 4− 22 . 8598 x 3+ 149 . 7122 x 2− 271 . 5277 x − 1 . 8598 12 . 8623; 6 . 9596; 3 . 0447; − 0 . 0068 100 x 4 − 22 . 8643 x 3 + 149 . 8248 x 2 − 272 . 3382 x − 0 . 1937 12 . 8621; 6 . 9593; 3 . 0437; − 0 . 0007 1000 x 4− 22 . 8648 x 3+ 149 . 8365 x 2− 272 . 4230 x − 0 . 0195 12 . 8620; 6 . 9592; 3 . 0436; − 0 . 0001 10 4 x 4− 22 . 8649 x 3+ 149 . 8377 x 2− 272 . 4315 x 12 . 8620; 6 . 9592; 3 . 0436; 0 T able 1. M 1 = 1; M 0 ≈ 10 4.
62 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ M 1 eL α(4 x ) M 0 = 1 α = 3 Zeros 0 . 1 x 4− 24 . 3286 x 3+ 175 . 7100 x 2− 388 . 7389 x + 103 . 0158 13 . 2109; 7 . 3279; 3 . 4844; 0 . 3054 0 . 5 x 4 − 23 . 5200 x 3 + 161 . 2800 x 2 − 322 . 5600 x + 40 . 3200 13 . 0123; 7 . 1213; 3 . 2526; 0 . 1338 0 . 8574 x 4− 22 . 9997 x 3+ 151 . 9942 x 2− 279 . 9734 x 12 . 8989; 6 . 9999; 3 . 1009; 0 1 x 4− 22 . 8285 x 3+ 148 . 9393 x 2− 265 . 9631 x − 13 . 2982 12 . 8638; 6 . 9621; 3 . 0512; − 0 . 0487 10 x 4 − 20 . 3604 x 3 + 104 . 8934 x 2 − 63 . 9594 x − 204 . 6701 12 . 4594; 6 . 5268; 2 . 4159; − 1 . 0418 100 x 4− 19 . 6699 x 3 + 92 . 5699 x 2− 7 . 4413 x − 258 . 2135 12 . 3734; 6 . 4375; 2 . 2805; − 1 . 4215 1000 x 4− 19 . 5882 x 3 + 91 . 1123 x 2− 0 . 7565 x − 264 . 5465 12 . 3639; 6 . 4277; 2 . 2658; − 1 . 4692 10 4 x 4 − 19 . 5799 x 3 + 90 . 9639 x 2 − 0 . 0758 x − 265 . 1914 12 . 3629; 6 . 4267; 2 . 2643; − 1 . 4740 T able 2. M 0 = 1; M 0 ≈ 0 . 8574.
M 0 eL α(4 x ) M 1 = 1 α = − 0 . 5 Zeros 0 . 1 x 4 − 11 . 4932 x 3 + 28 . 7232 x 2 − 4 . 5111 x − 4 . 0701 7 . 9670; 3 . 2871; 0 . 5315; − 0 . 2924 0 . 5 x 4 − 11 . 4652 x 3 + 28 . 5699 x 2 − 4 . 7602 x − 3 . 2078 7 . 9556; 3 . 2737; 0 . 4882; − 0 . 2523 1 x 4− 11 . 4434 x 3+ 28 . 4505 x 2− 4 . 9542 x − 2 . 5361 7 . 9467; 3 . 2633; 0 . 4504; − 0 . 2171 10 x 4− 11 . 3783 x 3+ 28 . 0941 x 2− 5 . 5331 x − 0 . 5318 7 . 9206; 3 . 2325; 0 . 2955; − 0 . 0703 100 x 4 − 11 . 3630 x 3 + 28 . 0102 x 2 − 5 . 6695 x − 0 . 0597 7 . 9146; 3 . 2253; 0 . 2332; − 0 . 0100 1000 x 4− 11 . 3613 x 3+ 28 . 0006 x 2− 5 . 6850 x − 0 . 0060 7 . 9139; 3 . 2245; 0 . 2239; − 0 . 0011 10 5 x 4− 11 . 3611 x 3+ 27 . 9996 x 2− 5 . 6867 x 7 . 9138; 3 . 2244; 0 . 2229; 0 T able 3. M 1 = 1; M 0 ≈ 10 5.
64 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ M 1 eL α(4 x ) M 0 = 1 α = − 0 . 5 Zeros 0 . 1 x 4− 12 . 5752 x 3+ 38 . 9473 x 2− 24 . 9535 x + 0 . 1805 8 . 1928; 3 . 5196; 0 . 8555; 0 . 0073 0 . 1122 x 4 − 12 . 5017 x 3 + 38 . 2656 x 2 − 23 . 6548 x 8 . 1746; 3 . 5004; 0 . 8264; 0 0 . 5 x 4− 11 . 6684 x 3+ 30 . 5375 x 2− 8 . 9306 x − 1 . 9960 7 . 9904; 3 . 3078; 0 . 5164; − 0 . 1462 1 x 4− 11 . 4434 x 3+ 28 . 4505 x 2− 4 . 9542 x − 2 . 5361 7 . 9467; 3 . 2633; 0 . 4504; − 0 . 2171 10 x 4 − 11 . 1941 x 3 + 26 . 1387 x 2 − 0 . 5496 x − 3 . 1344 7 . 9010; 3 . 2176; 0 . 3909; − 0 . 3154 100 x 4− 11 . 1661 x 3+ 25 . 8794 x 2− 0 . 0556 x − 3 . 2015 7 . 8960; 3 . 2127; 0 . 3851; − 0 . 3277 1000 x 4− 11 . 1633 x 3+ 25 . 8532 x 2− 0 . 0056 x − 3 . 2083 7 . 8955; 3 . 2122; 0 . 3846; − 0 . 3289 10 4 x 4 − 11 . 1630 x 3 + 25 . 8505 x 2 − 0 . 0006 x − 3 . 2090 7 . 8955; 3 . 2121; 0 . 3845; − 0 . 3291 T able 4. M 0 = 1; M 0 ≈ 0 . 1122.
Acknowledgements
The authors thank the valuable comments from the referees. They greatly con-tributed to improve the presentation of the manuscript. The third and fourth authors were supported by Promep and Conacyt.
References
[1] M. Alfaro, G. L´opez, and M. L. Rezola, Some Properties of Zeros of Sobolev-Type Orthogonal Polynomials, Journal of Computational and Ap-plied Mathematics69(1996), 171–179.
[2] G. E. Andrews, R. Askey, and R. Roy,Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, UK, 1999.
[3] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, USA, 1978.
[4] H. Due˜nas and F Marcell´an,The Laguerre-SoboleV-Type Orthogonal Poly-nomials, Journal of Approximation Theory162(2010), 421–440.
[5] H. Due˜nas and F. Marcell´an,The Laguerre-Sobolev-Type Orthogonal Poly-nomials. Holonomic Equation and Electrostatic Interpretation, Rocky Mount. Jour. Math41(2011), no. 1, 95–131.
[6] F. Gr¨unbaum, Variations on a Theme of Heine and Stieltjes: An Elec-trostatic Interpretation of the Zeros of Certain Polynomials, Journal of Comp. and App. Math.99(1998), 189–194.
[7] Mourad E. H. Ismail,An Electrostatic Model for Zeros of General Orthog-onal Polynomials, Pacific J. Math193(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, Cambridge University Press, Cambridge, UK, 2005.
[10] R. Koekoek, Generalization of the Classical Laguerre Polynomials and some Q-Analogues, Doctoral dissertation, Techn. Univ. of Delft, Delft, Netherlands, 1990.
[11] F. Marcell´an, A. Mart´ınez-Finkelshtein, and P. Mart´ınez-Gonz´alez, 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.
66 HERBERT DUE ˜NAS, JUAN C. GARC´ıA, LUIS E. GARZA & ALEJANDRO RAM´ıREZ
[12] G. Szeg¨o, Orthogonal Polynomials, Colloquium Publications - American Mathematical Society, vol. 23, American Mathematical Society, Provi-dence, USA, 4th ed., 1975.
(Recibido en agosto de 2012. Aceptado en enero de 2013)
Departamento de Matem´aticas Universidad Nacional de Colombia Facultad de Ciencias Carrera 30, calle 45 Bogot´a, Colombia e-mail: [email protected] e-mail: [email protected] Facultad de Ciencias Universidad de Colima Bernal D´ıaz del Castillo No. 340 Colonia Villas San Sebasti´an C.P. 28045 Colima, Colima, M´exico
e-mail:[email protected] e-mail:[email protected]
