Zeros of orthogonal polynomials generated by canonical perturbations of measures

Applied Mathematics and Computation 218 (2012) 7109–7127

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Zeros of orthogonal polynomials generated by canonical perturbations of measures q Edmundo J. Huertas a,⇑, Francisco Marcellán a, Fernando R. Rafaeli b a b

Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III, Leganés, Madrid, Spain Departamento de Matemática, Estatística e Computação, FCT, Universidade Estadual Paulista – Unesp, Brazil

a r t i c l e

i n f o

Keywords: Orthogonal polynomials Interlacing Monotonicity Asymptotic behavior Electrostatics interpretation

a b s t r a c t In this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the Uvarov perturbation of a positive Borel measure dlðxÞ. When the measure is semiclassical, then its electrostatic interpretation is given. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction and statement of the main results This paper deals with the behavior of the zeros of the sequence of monic polynomials fpn ðk; c; xÞgnP0 orthogonal with respect to the Uvarov perturbation dlðk; c; xÞ ¼ dlðxÞ þ kdðx  cÞ, where dlðxÞ is a positive Borel measure supported in a finite or infinite interval ða; bÞ; dðx  cÞ is the Dirac delta functional at c, with c R ða; bÞ, and k is a nonnegative real number. In other words this means that this sequence of polynomials is orthogonal with respect to the inner product

hp; qik;c ¼

Z

b

pðxÞqðxÞ dlðxÞ þ kpðcÞqðcÞ:

ð1Þ

a

Let xn;k ðk; cÞ; k ¼ 1; . . . ; n, be the zeros of pn ðk; c; xÞ. If k is a nonnegative real number then dlðk; c; xÞ is a positive measure, and, as a consequence, the zeros of pn ðk; c; xÞ are real, simple, and lie in (c, b) (resp. in (a, c)) if c 6 a (resp. if c P b), that is,

minfa; cg < xn;1 ðk; cÞ <    < xn;n ðk; cÞ < maxfb; cg: In particular, if k ¼ 0 we denote by xn;k :¼ xn;k ð0; cÞ the zeros of pn ðxÞ :¼ pn ð0; c; xÞ, the sequence of monic polynomials orthogonal with respect to the measure dlðxÞ. Now, some natural questions arise: Are there values of the parameter k for which the zeros of pn ðk; c; xÞ interlace with the zeros of pn ðxÞ? Are the zeros xn;k ðk; cÞ monotonic functions with respect to the parameter k? Do the zeros xn;k ðk; cÞ converge when k goes to infinity? If so, what the speed of convergence is? One of our main contributions regards the questions posed above. We provide an interlacing property as well as the monotonicity and asymptotic behaviour of the zeros of the polynomial pn ðk; c; xÞ with respect to k.

q This work has been done in the framework of a joint project of Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain and the Brazilian Science Foundation CAPES, Project CAPES/DGU 160/08. The work of the first and second authors has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, Grant MTM2009-12740-C03-01. The work of the third author has been supported by CNPq, FAPESP and FUNDUNESP. ⇑ Corresponding author. E-mail addresses: [email protected] (E.J. Huertas), [email protected] (F. Marcellán), [email protected] (F.R. Rafaeli).

0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.12.073

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Theorem 1. Let k > 0 and zn;1 ðcÞ; . . . ; zn;n ðcÞ be the zeros of the polynomial r n ðc; xÞ defined below. (i) If c 6 a, then

c < xn;1 ðk; cÞ < xn;1 < zn1;1 ðcÞ < xn;2 ðk; cÞ < xn;2 <    < zn1;n1 ðcÞ < xn;n ðk; cÞ < xn;n : Moreover, each xn;k ðk; cÞ is a decreasing function of k and, for each k ¼ 1; . . . ; n  1,

lim xn;1 ðk; cÞ ¼ c;

k!1

lim xn;kþ1 ðk; cÞ ¼ zn1;k ðcÞ;

k!1

as well as

lim k½xn;1 ðk; cÞ  c ¼

k!1

pn ðcÞ ; K n1 ðc; cÞr n1 ðc; cÞ

lim k½xn;kþ1 ðkÞ  zn1;k ðcÞ ¼

k!1

pn ðzn1;k ðcÞÞ : K n1 ðc; cÞðzn1;k ðcÞ  cÞ½r n1 ðc; xÞ0x¼zn1;k ðcÞ

ð2Þ

(ii) If c P b, then

xn;1 < xn;1 ðk; cÞ < zn1;1 ðcÞ <    < xn;n1 < xn;n1 ðk; cÞ < zn1;n1 ðcÞ < xn;n < xn;n ðk; cÞ < c: Moreover, each xn;k ðk; cÞ is an increasing function of k and, for each k ¼ 1; . . . ; n  1,

lim xn;n ðk; cÞ ¼ c;

k!1

lim xn;k ðk; cÞ ¼ zn1;k ðcÞ

k!1

and

lim k½c  xn;n ðk; cÞ ¼

k!1

pn ðcÞ ; K n1 ðc; cÞr n1 ðc; cÞ

lim k½zn1;k ðcÞ  xn;k ðk; cÞ ¼

k!1

pn ðzn1;k ðcÞÞ : K n1 ðc; cÞðzn1;k ðcÞ  cÞ½rn1 ðc; xÞ0x¼zn1;k ðcÞ

Note that the mass point c attracts one zero of pn ðk; c; xÞ, that is, when k goes to infinity, it captures either the smallest or the largest zero, according to the location of the point c with respect to the interval (a, b). In addition, when either c < a or c > b, at most one of the zeros of pn ðk; c; xÞ is located outside (a, b). We point out that Theorem 1 is general in two aspects and uses new approaches to the analyses of zeros: dlðxÞ is any positive Borel measure and c is any value outside (a, b). Some particular cases of these polynomials appear in the seminal papers by H.L. Krall [25] and A.M. Krall [24] devoted to the spectral analysis of fourth order linear differential operators with polynomial coefficients. Koornwinder [23] analyzed a general situation for Jacobi weights when two masses are added at the end points of the interval [1, 1]. Later on, in [15], Krall–Hermite and Krall–Bessel polynomials are studied in the framework of Darboux transformations. In [22] analytic properties of orthogonal polynomials with respect to a perturbation of the Laguerre weight when a mass is added at x ¼ 0 are considered. In [28], the holonomic equation for such perturbations when the mass point is located in the negative real semi-axis is deduced. In the framework of the spectral theory of higher order linear differential operators, in [20,21] the authors obtain infinite order differential operators such that the Krall-Laguerre and Krall-Jacobi are their eigenfunctions, respectively. In particular, for some choices of the parameters of Laguerre and Jacobi weights they prove that the differential operator has a finite order. On the other hand, in [26] the authors deduced the invariance of the semiclassical character of semiclassical linear functionals under Uvarov transformations independently of the location of the mass point. All the above questions concerning the behaviour of the zeros of the polynomials pn ðk; c; xÞ were answered for two important and particular cases in [6,7]. The authors considered the cases when dlðxÞ ¼ xa ex dx with ða; bÞ ¼ ð0; 1Þ and c ¼ 0, and dlðxÞ ¼ ð1  xÞa ð1 þ xÞb dx with ða; bÞ ¼ ð1; 1Þ and c ¼ 1, respectively. We also provide the second order linear differential equation that the polynomial pn ðk; c; xÞ satisfies when the measure dlðxÞ in (1) is semiclassical (for definition of a semiclassical measure see [29]). This is the main tool for the electrostatic interpretation of zeros. Theorem 2. The monic orthogonal polynomial sequence fpn ðk; c; xÞgnP0 satisfies the holonomic equation (second order linear differential equation)

Aðx; nÞðpn ðk; c; xÞÞ00 þ Bðx; nÞðpn ðk; c; xÞÞ0 þ Cðx; nÞpn ðk; c; xÞ ¼ 0;

ð3Þ

E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

7111

where

Aðx; nÞ ¼

cn ½/ðxÞ2 ; e nÞ e nÞ  cn Aðx; Bðx; h i e nÞ þ cn ð/0 ðxÞ  Aðx; nÞÞ /ðxÞ Bðx; nÞ  Bðx;

 0 e nÞ e nÞ  cn Aðx; cn /ðxÞ2 Bðx; ;  Bðx; nÞ ¼  2 e nÞ e nÞ  cn Aðx; Bðx; e nÞ e nÞ  cn Aðx; Bðx; ! e nÞ e nÞ e nÞ  Bðx; nÞ Aðx; Bðx; Aðx; nÞ Bðx; Cðx; nÞ ¼  /ðxÞD : e nÞ e nÞ e nÞ  cn Aðx; e nÞ  cn Aðx; Bðx; Bðx; In order to justify our approach, we must point out that the behavior of the zeros of orthogonal polynomials has been extensively studied because of their applications in many areas of physics and engineering. First, the zeros of orthogonal polynomials are the nodes of the Gaussian quadrature rules and also play an important role in some of their extensions like Gauss–Radau, Gauss– Lobatto, and Gauss–Kronrod rules, among others (see [5,12,30]). Second, the zeros of classical orthogonal polynomials are the electrostatic equilibrium points of positive unit charges interacting according to a logarithmic }’s book [38, Section 6.7], and some recent potential under the action of an external field, see Stieltjes’ papers [34–37], Szego works like Dimitrov and Van Assche [8], Grunbaum [13,14], Ismail [17], and Marcellán et al. [27] among others. Third, in a more general framework, the counting measure of zeros weakly converges to the equilibrium measure associated with a logarithmic potential (see [33]). Fourth, zeros of orthogonal polynomials are used in collocation methods for boundary value problems of second order linear differential operators (see [2]). Fifth, global properties of zeros of orthogonal polynomials can be analyzed when they satisfy second order differential equations with polynomial coefficients using the WKB method (see [1]). Finally, zeros of orthogonal polynomials are eigenvalues of Jacobi matrices and its role in Numerical Linear Algebra is very well known. The structure of the manuscript is as follows. In Section 2 we prove Theorem 1. It follows from the Christoffel formula, from connections formula for the perturbed polynomials in terms of the initial ones and from a lemma concerning the behavior of the zeros of a linear combination of two polynomials. In addition, we obtain new connection formulas for orthogonal polynomials obtained from Uvarov and Christoffel transformations and some results about their zeros. In Section 3, we check these results for the Jacobi-type and Laguerre-type orthogonal polynomials introduced by Koornwinder [23]. In Section 4 we obtain the holonomic equations that the polynomials pn ðk; c; xÞ satisfy using an alternative approach to the standard ones. We focus our attention on the electrostatic interpretation of the zeros as equilibrium points in a logarithmic potential interaction of positive unit charges under the presence of an external field. We analyze such an equilibrium problem when the mass point is located either on the boundary or in the exterior of the support of the measure, respectively, for the Laguerre and Jacobi weights transformations. 2. Proof of Theorem 1 In this section, we prove Theorem 1 and present some new results. Proof of Theorem 1. Le yn;k ðcÞ be the zeros of the monic polynomials qn ðc; xÞ orthogonal with respect to the perturbed measure

dl1 ðc; xÞ ¼ jx  cj dlðxÞ; where c R ða; bÞ. This perturbation is the so-called Christoffel perturbation (see [41,40]). It is well known that qn ðc; xÞ is the monic kernel polynomial which can be represented as (see [5, (7.3)])

qn ðc; xÞ ¼

  kpn k2l 1 p ðcÞ pnþ1 ðxÞ  nþ1 pn ðxÞ ¼ K n ðc; xÞ; xc pn ðcÞ pn ðcÞ

ð4Þ

where K n ðc; xÞ is the nth kernel polynomial defined by

K n ðc; xÞ ¼

n X pj ðcÞpj ðxÞ j¼0

kpj k2l

:

ð5Þ

In Chihara’s book [2, Theorem 7.2] we found the following interlacing property involving the zeros of qn ðc; xÞ; pnþ1 ðxÞ and pn ðxÞ:  If c 6 a, then

xnþ1;1 < xn;1 < yn;1 ðcÞ < xnþ1;2 <    < xn;n < yn;n ðcÞ < xnþ1;nþ1 :  If c P b, then

xnþ1;1 < yn;1 ðcÞ < xn;1 <    < xnþ1;n < yn;n ðcÞ < xn;n < xnþ1;nþ1 :

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We introduce the monic polynomials r n ðc; xÞ orthogonal with respect to the measure

dl2 ðc; xÞ ¼ jx  cj dl1 ðc; xÞ ¼ ðx  cÞ2 dlðxÞ: Using (4) we deduce that

rn ðc; xÞ ¼

1 q ðc; cÞ 1 ½q ðc; xÞ  nþ1 ½pnþ2 ðxÞ  dn pnþ1 ðxÞ þ en pn ðxÞ; q ðc; xÞ ¼ x  c nþ1 qn ðc; cÞ n ðx  cÞ2

ð6Þ

where

dn ¼

pnþ2 ðcÞ qnþ1 ðc; cÞ pnþ2 ðcÞ þ pn ðcÞ þ ¼ en ; pnþ1 ðcÞ qn ðc; cÞ pnþ1 ðcÞ

en ¼

qnþ1 ðc; cÞ pnþ1 ðcÞ kpnþ1 kl K nþ1 ðc; cÞ ¼ > 0: qn ðc; cÞ pn ðcÞ kpn k2l K n ðc; cÞ

2

Notice that rn ðc; cÞ – 0. If we denote by zn;k ðcÞ the zeros of r n ðc; xÞ, then using the three term recurrence relation

pnþ1 ðxÞ ¼ ðx  bn Þpn ðxÞ  cn pn1 ðxÞ; where

bn ¼

hxpn ; pn il kpn k2l

;

n P 0;

and cn ¼

kpn k2l kpn1 k2l

> 0;

n P 1;

in (6) we obtain

rn ðc; xÞ ¼

1 ðx  cÞ2

½ðx  bnþ1  dn Þpnþ1 ðxÞ þ ðen  cnþ1 Þpn ðxÞ:

ð7Þ

On the other hand,

en  cnþ1 ¼

  kpnþ1 k2l K nþ1 ðc; cÞ  1 > 0: K n ðc; cÞ kpn k2l

ð8Þ

Thus, evaluating r n ðc; xÞ at the zeros xnþ1;k , from (7) and (8),

Sign½r n ðc; xnþ1;k Þ ¼ Sign½pn ðxnþ1;k Þ;

k ¼ 1; . . . ; n þ 1:

Since the zeros of pnþ1 ðxÞ and pn ðxÞ interlace, we conclude that h Theorem 3. The inequalities

xnþ1;1 < zn;1 ðcÞ < xnþ1;2 < zn;2 ðcÞ <    < xnþ1;n < zn;n ðcÞ < xnþ1;nþ1 ;

ð9Þ

hold for every n 2 N. In [1, (8)] (see also [39]) the authors show that

pn ðk; c; xÞ ¼ pn ðxÞ 

kpn ðcÞ K n1 ðc; xÞ: 1 þ kK n1 ðc; cÞ

ð10Þ

In [16] another connection formula was obtained. Using the similar idea as in Proposition 4 in [16] we obtain ^n ðk; c; xÞ, with the normalization p ^n ðk; c; xÞ ¼ kn pn ðk; c; xÞ, can be represented Theorem 4 (Connection formula). The polynomials p as

^n ðk; c; xÞ ¼ pn ðxÞ þ kK n1 ðc; cÞðx  cÞrn1 ðc; xÞ; p

ð11Þ

where kn ¼ 1 þ kK n1 ðc; cÞ. Using the interlacing property (9) and the connection formula (11), we get

^n ðk; c; xn;k Þ ¼ Sign½kðxn;k  cÞrn1 ðc; xÞ; Sign½p

k ¼ 1; . . . ; n

and

^n ðk; c; zn1;k ðcÞÞ ¼ Sign½pn ðzn1;k ðcÞÞ; Sign½p

k ¼ 1; . . . ; n  1;

which yield the inequalities stated in Theorem 1. It remains to show the monotonicity, asymptotics and the speed of the convergence of the zeros xn;k ðk; cÞ with respect to k. Indeed, it follows from the technique developed in [3, Lemma 1] and

E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

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[7, Lemmas 1 and 2] (see also [31, Theorem 3.9]) concerning the zeros of a linear combination of two polynomials with interlacing zeros. We also derive a representation for the monic polynomial pn ðk; c; xÞ as a combination of two Christoffel polynomials it will be very useful for obtain the holomonic equation for pn ðk; c; xÞ. Theorem 5. The monic polynomials pn ðk; c; xÞ can be also represented as

pn ðk; c; xÞ ¼ qn ðc; xÞ þ cn qn1 ðc; xÞ;

ð12Þ

where

cn ¼

1 þ kK n ðc; cÞ pn1 ðcÞ c 1 þ kK n1 ðc; cÞ pn ðcÞ n

and cn ¼

kpn k2l kpn1 k2l

:

ð13Þ

Using (5), we obtain

pn ðxÞ ¼

kpn k2l pn ðcÞ

½K n ðc; xÞ  K n1 ðc; xÞ:

ð14Þ

From (4) and (14), we have

pn ðxÞ ¼ qn ðc; xÞ  cn

pn1 ðcÞ q ðc; xÞ: pn ðcÞ n1

ð15Þ

Therefore, substituting (4) and (15) in (10) we get (12). In the following we deduce the value k0 of the mass such that for k > k0 one of the zeros of pn ðk; c; xÞ is located outside (a, b). Corollary 1. Let k > 0. (i) If c < a, then the smallest zero xn;1 ðk; cÞ satisfies

xn;1 ðk; cÞ > a; xn;1 ðk; cÞ ¼ a;

for k < k0 ; for k ¼ k0 ;

xn;1 ðk; cÞ < a;

for k > k0 ;

where

k0 ¼ k0 ðn; a; cÞ ¼

pn ðaÞ > 0: K n1 ðc; cÞða  cÞr n1 ðc; aÞ

(ii) If c > b, then the largest zero xn;n ðk; cÞ satisfies

xn;n ðk; cÞ < b;

for k < k0 ;

xn;n ðk; cÞ ¼ b;

for k ¼ k0 ;

xn;n ðk; cÞ > b;

for k > k0 ;

where k0 ¼ k0 ðn; b; cÞ. The proofs are a consequence of the connection formula (11). 3. Application to classical measures 3.1. Jacobi type (Jacobi–Koornwinder) orthogonal polynomials ða;bÞ

Let fpn

ðxÞgnP0 be the monic Jacobi polynomial sequence which is orthogonal with respect to the measure dla;b ðxÞ ¼

a

ð1  xÞ ð1 þ xÞb dx; a; b > 1, supported on ð1; 1Þ. We consider the following Uvarov perturbations of dla;b ðxÞ where either a ¼ 1 or a ¼ 1, and k P 0.

dlðk; 1; xÞ ¼ dla;b ðxÞ þ kdðx þ 1Þ;

ð16Þ

dlðk; 1; xÞ ¼ dla;b ðxÞ þ kdðx  1Þ:

ð17Þ

Such orthogonal polynomials were first studied by Koornwinder (see [23]), in 1984. There, he adds simultaneously two Dirac delta functions at the end points x ¼ 1 and x ¼ 1, that is,

dlM;N ðxÞ ¼ dla;b ðxÞ þ Mdðx þ 1Þ þ Ndðx  1Þ:

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^ðna;bÞ ðk; 1; xÞgnP0 and fp ^ðna;bÞ ðk; 1; xÞgnP0 the sequences of orthogonal polynomials with respect (16) and (17), with Denote by fp the normalization introduced in Theorem 2, respectively. Then, the connection formulas are ða;bþ2Þ ^ðna;bÞ ðk; 1; xÞ ¼ pðna;bÞ ðxÞ þ kK n1 ð1; 1Þðx þ 1Þpn1 ðxÞ p

and aþ2;bÞ ^ðna;bÞ ðk; 1; xÞ ¼ pnða;bÞ ðxÞ þ kK n1 ð1; 1Þðx  1Þpðn1 ðxÞ: p

It is straightforward to see that

K n1 ð1; 1Þ ¼

1 Cðn þ b þ 1ÞCðn þ a þ b þ 1Þ 2aþbþ1 CðnÞCðb þ 1ÞCðb þ 2ÞCðn þ aÞ

and

K n1 ð1; 1Þ ¼

1 Cðn þ a þ 1ÞCðn þ a þ b þ 1Þ : 2aþbþ1 CðnÞCða þ 1ÞCða þ 2ÞCðn þ bÞ

^nða;bÞ ðk; 1; xÞ and Recently, several authors [1,7,11] have been contributed to the analysis of the behavior of the zeros of p ða;bÞ ^ pn ðk; 1; xÞ. ^nða;bÞ ðk; 1; xÞ and pðna;bÞ ðxÞ, respecLet us denote by xn;k ða; b; kÞ :¼ xn;k ða; b; 1; kÞ and xn;k ða; bÞ; k ¼ 1; . . . ; n, the zeros of p tively, in an increasing order. Then, applying Theorem 1, we obtain. Theorem 6 [7]. The inequalities

1 < xn;1 ða; b; kÞ < xn;1 ða; bÞ < xn1;1 ða; b þ 2Þ < xn;2 ða; b; kÞ < xn;2 ða; bÞ <    < xn1;n1 ða; b þ 2Þ < xn;n ða; b; kÞ < xn;n ða; bÞ; hold for every a; b > 1. Moreover, each xn;k ða; b; kÞ is a decreasing function of k and, for each k ¼ 1; . . . ; n  1,

lim xn;1 ða; b; kÞ ¼ 1;

k!1

lim xn;kþ1 ða; b; kÞ ¼ xn1;k ða; b þ 2Þ

k!1

and

lim k½xn;1 ða; b; kÞ þ 1 ¼ hn ða; bÞ;

k!1

lim k½xn;kþ1 ða; b; kÞ  xn1;k ða; b þ 2Þ ¼

k!1

½1  xn1;k ða; b þ 2Þhn ða; bÞ ; 2ðb þ 2Þ

where

hn ða; bÞ ¼

2aþbþ2 CðnÞCðb þ 2ÞCðb þ 3ÞCðn þ aÞ : Cðn þ b þ 2ÞCðn þ a þ b þ 2Þ

From (2) ða;bÞ

lim k½xn;1 ða; b; kÞ þ 1 ¼

k!1

pn

ð1Þ ða;bþ2Þ

K n1 ð1; 1Þpn1

ð1Þ

:

Since

pðna;bÞ ð1Þ ¼

ð1Þn 2n Cðn þ b þ 1ÞCðn þ a þ b þ 1Þ Cðb þ 1ÞCð2n þ a þ b þ 1Þ

and

K n1 ð1; 1Þ ¼

1 Cðn þ b þ 1ÞCðn þ a þ b þ 1Þ ; 2aþbþ1 CðnÞCðb þ 1ÞCðb þ 2ÞCðn þ aÞ

we obtain ða;bÞ

pn

ð1Þ

ða;bþ2Þ K n1 ð1; 1Þpn1 ð1Þ

¼

2aþbþ2 CðnÞCðb þ 2ÞCðb þ 3ÞCðn þ aÞ ¼ hn ða; bÞ: Cðn þ b þ 2ÞCðn þ a þ b þ 2Þ

It also follows from (2) that ða;bÞ

lim k½xn;kþ1 ða; b; kÞ  xn1;k ða; b þ 2Þ ¼

k!1

pn

ðxn1;k ða; b þ 2ÞÞ ða;bþ2Þ

K n1 ð1; 1Þðxn1;k ða; b þ 2Þ þ 1Þ½pn1

ðxÞ0 jx¼xn1;k ða;bþ2Þ

:

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On the other hand, from ða;bþ2Þ

nðn þ aÞð1 þ xÞpn1

ðxÞ ¼ nðn þ a þ b þ 1Þpðna;bÞ ðxÞ þ ðb þ 1Þð1  xÞ½pðna;bÞ ðxÞ0 ;

we derive

 nðn þ a þ b þ 1Þpðna;bÞ ðxn1;k ða; b þ 2ÞÞ ¼ ðb þ 1Þð1  xn1;k ða; b þ 2ÞÞ½pðna;bÞ ðxÞ0 x¼x

n1;k ða;bþ2Þ

;

as well as

 nðn þ aÞð1 þ xn1;k ða; b þ 2ÞÞ½pðna;bÞ ðxÞ0 x¼x

n1;k ða;bþ2Þ

 ¼ ½nðn þ a þ b þ 1Þ  ðb þ 1Þ½pðna;bÞ ðxÞ0 x¼x

n1;k ða;bþ2Þ

 þ ðb þ 1Þð1  xn1;k ða; b þ 2ÞÞ½pðna;bÞ ðxÞ00 x¼x

n1;k ða;bþ2Þ

:

Using the last two equalities and the differential equation for the Jacobi polynomials

ð1  x2 Þ½pðna;bÞ ðxÞ00 þ ½b  a  ða þ b þ 1Þx½pnða;bÞ ðxÞ0 þ nðn þ a þ b þ 1Þpðna;bÞ ðxÞ ¼ 0; we obtain

ð1 þ xn1;k ða; b þ 2ÞÞ½pðna;bþ2Þ ðxÞ0 jx¼xn1;k ða;bþ2Þ ¼

ðn þ b þ 1Þðn þ a þ b þ 1Þ ða;bÞ p ðxn1;k ða; b þ 2ÞÞ: ðb þ 1Þð1  xn1;k ða; b þ 2ÞÞ n

Therefore, ða;bÞ

lim k½xn;kþ1 ða; b; kÞ  xn1;k ða; b þ 2Þ ¼

k!1

pn

ðxn1;k ða; b þ 2ÞÞ ða;bþ2Þ

K n1 ð1; 1Þðxn1;k ða; b þ 2Þ þ 1Þ½pn1

  ðxÞ0 

x¼xn1;k ða;bþ2Þ

½1  xn1;k ða; b þ 2Þhn ða; bÞ : ¼ 2ðb þ 2Þ ^ðna;bÞ ðk; 1; xÞ. Then Let xn;k ða; b; kÞ :¼ xn;k ða; b; 1; kÞ be the zeros of p Theorem 7 [7]. The inequalities

xn;1 ða; bÞ < xn;1 ða; b; kÞ < xn1;1 ða þ 2; bÞ <    < xn;n1 ða; bÞ < xn;n1 ða; b; kÞ < xn1;n1 ða þ 2; bÞ < xn;n ða; bÞ < xn;n ða; b; kÞ < 1; hold for every a; b > 1. Moreover, each xn;k ða; b; kÞ is an increasing function of k and, for each k ¼ 1; . . . ; n  1,

lim xn;n ða; b; kÞ ¼ 1;

k!1

lim xn;k ða; b; kÞ ¼ xn1;k ða þ 2; bÞ

k!1

and

lim k½1  xn;n ða; b; kÞ ¼ g n ða; bÞ;

k!1

lim k½xn1;k ða þ 2; bÞ  xn;k ða; b; kÞ ¼

k!1

½1 þ xn1;k ða þ 2; bÞg n ða; bÞ ; 2ða þ 2Þ

where

g n ða; bÞ ¼

2aþbþ2 CðnÞCða þ 2ÞCða þ 3ÞCðn þ bÞ : Cðn þ a þ 2ÞCðn þ a þ b þ 2Þ

We can proceed as in the proof of Theorem 6. We only observe that

nþb aþ1 ðaþ2;bÞ ðx  1Þpn1 ðxÞ ¼ pðna;bÞ ðxÞ  ð1 þ xÞ½pðna;bÞ ðxÞ0 : 2n nðn þ a þ b þ 1Þ ða;bÞ

^3 ðk þ e; 1; xÞ, for a ¼ b ¼ 0 and some values of e > 0 appear in Fig. 1 To illustrate the results of Theorem 7, the graphs of p ^ð3a;bÞ ðk; 1; xÞ as a function of the mass k is clarified. where the monotonicity of the zeros of p ^3ða;bÞ ðk; 1; xÞ, with a ¼ b ¼ 0, for several choices of k. Notice that the largest zero conIn Table 1 we describe the zeros of p ð2;0Þ verges to 1 and the other two zeros converge to the zeros of the Jacobi polynomial p2 ðxÞ, that is, they converge to x2;1 ð2; 0Þ ¼ 0:75497 and x2;2 ð4Þ ¼ 0:0883037. Also note that all the zeros increase when k increase. 3.2. Laguerre type (Laguerre–Koornwinder) orthogonal polynomials ðaÞ

Let fpn ðxÞgnP0 be the monic Laguerre polynomial which are orthogonal with respect to the measure dlðxÞ ¼ xa ex dx; a > 1, supported on ð0; þ1Þ. We will consider the Uvarov perturbation on dla ðxÞ with c ¼ 0

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1

1

1

x

1 ^3ða;bÞ ðk þ ; 1; xÞ for some values of e. Fig. 1. The graphs of p

Table 1 ^ð3a;bÞ ðk; 1; xÞ for some values of k. Zeros of p k

x3;1 ð0; 0; kÞ

x3;2 ð0; 0; kÞ

x3;3 ð0; 0; kÞ

0 1 10 100 1000

0.774597 0.757872 0.755305 0.755004 0.754974

0 0.0753429 0.0868168 0.0881528 0.0882886

0.774597 0.955257 0.994575 0.999446 0.999944

dlðk; 0; xÞ ¼ dlðxÞ þ kdðxÞ; ðaÞ pn ðk; xÞ

k P 0:

ð18Þ

ðaÞ pn ðk; 0; xÞ

The polynomial :¼ orthogonal with respect to (18) was also obtained by Koornwinder [23] as a special limit case of the Jacobi–Koornwinder (Jacobi type) orthogonal polynomial. Analytic properties of these polynomials have been studied in the last years (see [1,6,10,22], among others). The connection formula (11) reads aþ2Þ ^ðnaÞ ðk; xÞ ¼ pðnaÞ ðxÞ þ kK n1 ð0; 0Þxpðn1 ðxÞ; p

where

K n1 ð0; 0Þ ¼

Cðn þ a þ 1Þ : CðnÞCða þ 1ÞCða þ 2Þ

Now, we will analyze the behavior of their zeros. Let denote by xn;k ða; kÞ and xn;k ðaÞ; k ¼ 1; . . . ; n, the zeros of the Laguerre type and the classical Laguerre orthogonal polynomials, respectively. Applying the results of Theorem 1 we obtain Theorem 8 [6]. The inequalities

0 < xn;1 ða; kÞ < xn;1 ðaÞ < xn1;1 ða þ 2Þ < xn;2 ða; kÞ < xn;2 ðaÞ <    < xn1;n1 ða þ 2Þ < xn;n ða; kÞ < xn;n ðaÞ; hold for every a > 1. Moreover, each xn;k ða; kÞ is a decreasing function of k and, for each k ¼ 1; . . . ; n  1,

lim xn;1 ða; kÞ ¼ 0;

k!1

lim xn;kþ1 ða; kÞ ¼ xn1;k ða þ 2Þ;

k!1

as well as

lim kxn;1 ða; kÞ ¼ g n ðaÞ;

k!1

lim k½xn;kþ1 ða; kÞ  xn1;k ða þ 2Þ ¼

k!1

g n ðaÞ ; aþ2

where

g n ðaÞ ¼

CðnÞCða þ 2ÞCða þ 3Þ : Cðn þ a þ 2Þ

From (2) ðaÞ

lim kxn;1 ða; kÞ ¼

k!1

pn ð0Þ ðaþ2Þ

K n1 ð0; 0Þpn1 ð0Þ

:

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Since

pðnaÞ ð0Þ ¼

ð1Þn Cðn þ a þ 1Þ Cða þ 1Þ

and K n1 ð0; 0Þ ¼

Cðn þ a þ 1Þ ; CðnÞCða þ 1ÞCða þ 2Þ

we obtain ðaÞ

pn ð0Þ ðaþ2Þ

K n1 ð0; 0Þpn1 ð0Þ

¼

CðnÞCða þ 2ÞCða þ 3Þ ¼ g n ðaÞ: Cðn þ a þ 2Þ

From (2) ðaÞ

lim k½xn;kþ1 ða; kÞ  xn1;k ða þ 2Þ ¼

k!1

pn ðxn1;k ða þ 2ÞÞ

  ðaþ2Þ K n1 ð0; 0Þxn1;k ða þ 2Þ½pn1 ðxÞ0 

:

x¼xn1;k ðaþ2Þ

On the other hand, It is easy to verify that ðaþ2Þ

xpn1 ðxÞ ¼ pnðaÞ ðxÞ þ

aþ1 n

½pðnaÞ ðxÞ0 :

Thus,

 ½pðnaÞ ðxÞ0 x¼x

n1;k ðaþ2Þ

¼

n

aþ1

pðnaÞ ðxn1;k ða þ 2ÞÞ

and

 xn1;k ða þ 2Þ½pðnaÞ ðxÞ0 x¼x

n1;k ðaþ2Þ

 ¼ ½pðnaÞ ðxÞ0 x¼x

n1;k ðaþ2Þ

þ

aþ1 n

 ½pðnaÞ ðxÞ00 x¼x

n1;k ðaþ2Þ

:

Using the last two equalities and the differential equation for the Laguerre polynomials

x½pðnaÞ ðxÞ00 þ ða þ 1  xÞ½pðnaÞ ðxÞ0 þ npðnaÞ ðxÞ ¼ 0; we obtain

 xn1;k ða þ 2Þ½pðnaÞ ðxÞ0 x¼x

n1;k ðaþ2Þ

¼

ðn þ a þ 1Þ ðaÞ pn ðxn1;k ða þ 2ÞÞ: aþ1

30

5

10

x

30 ðaÞ

Fig. 2. The graphs of p3 ðk þ e; xÞ for some values of e.

Table 2 ðaÞ Zeros of p3 ðk; xÞ for some values of k. k

x3;1 ð2; kÞ

x3;2 ð2; kÞ

x3;3 ð2; kÞ

0 1 10 100 1000

1.51739 0.321731 0.0390611 0.00399042 0.00039990

4.31158 3.64053 3.5604 3.55151 3.55061

9.17103 8.53774 8.45936 8.45049 8.44959

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Therefore, ðaÞ

lim k½xn;kþ1 ða; kÞ  xn1;k ða þ 2Þ ¼

k!1

pn ðxn1;k ða þ 2ÞÞ K n1 ð0; 0Þxn1;k ða þ



 ðaþ2Þ 2Þ½pn1 ðxÞ0  x¼xn1;k ðaþ2Þ

¼

CðnÞCða þ 2ÞCða þ 2Þ g n ðaÞ ¼ : Cðn þ a þ 2Þ aþ2

ðaÞ

To illustrate the results of Theorem 8 we enclose the graphs of p3 ðk þ e; xÞ, for a ¼ 2 and some values of e > 0. The Fig. 2 ðaÞ shows the monotonicity of the zeros of p3 ðk; xÞ as a function of the mass k. ðaÞ The Table 2 describes the zeros of p3 ðk; xÞ , with a ¼ 2, for several choices of k. Observe that the smallest zero converges ð4Þ to 0 and the other two zeros converge to the zeros of the Laguerre polynomial p2 ðxÞ, that is, they converge to x2;1 ð4Þ ¼ 3:55051 and x2;2 ð4Þ ¼ 8:44949. Note that all the zeros decrease when k increases. 4. Semiclassical orthogonal polynomials and spectral transformations We assume that dlðxÞ ¼ xðxÞ dx, where xðxÞ is a weight function supported on the real line. We can associate with xðxÞ an external potential tðxÞ such that xðxÞ ¼ expðtðxÞÞ. Notice that if tðxÞ is assumed to be differentiable on the support of dlðxÞ ¼ xðxÞ dx then

x0 ðxÞ ¼ t0 ðxÞ: xðxÞ If t0 ðxÞ is a rational function on (a, b), then the weight function xðxÞ is said to be semiclassical (see [29,32]). The linear functional u associated with xðxÞ,

hu; pðxÞi ¼

Z

b

pðxÞxðxÞ dx;

a

satisfies a distributional equation (which is known in the literature as Pearson equation)

DðrðxÞuÞ ¼ sðxÞu; where rðxÞ and sðxÞ are non-zero polynomials such that rðxÞ is monic and deg ðsðxÞÞ P 1. Notice that, in terms of the weight function, the above relation reads

x0 ðxÞ sðxÞ  r0 ðxÞ sðxÞ  r0 ðxÞ ¼ ; or; equivalently t0 ðxÞ ¼  : xðxÞ rðxÞ rðxÞ Let consider the linear functional u1 associated with the measure dl1 ðc; xÞ. In order to find the Pearson equation that u1 satisfies, we analyze two situations:  If

rðcÞ – 0, then Dððx  cÞrðxÞu1 Þ ¼ Dððx  cÞ2 rðxÞuÞ ¼ 2ðx  cÞrðxÞu þ ðx  cÞ2 DðrðxÞuÞ ¼ 2rðxÞu1 þ ðx  cÞ2 sðxÞu ¼ ½2rðxÞ þ ðx  cÞsðxÞu1 :

Thus,

Dð/ðxÞu1 Þ ¼ wðxÞu1 ; where

  /ðxÞ ¼ ðx  cÞrðxÞ;   wðxÞ ¼ 2rðxÞ þ ðx  cÞsðxÞ:  If

ð19Þ

rðcÞ ¼ 0, i.e., rðxÞ ¼ ðx  cÞr ðxÞ, then  ðxÞuÞ ¼ Dððx  cÞrðxÞuÞ ¼ rðxÞu þ ðx  cÞDðrðxÞuÞ ¼ rðxÞu þ ðx  cÞsðxÞu  ðxÞu1 Þ ¼ Dððx  cÞ2 r DðrðxÞu1 Þ ¼ Dððx  cÞr  ðxÞ þ sðxÞÞu1 : ¼ ðr

In this case,

Dð/ðxÞu1 Þ ¼ wðxÞu1 ; with

  /ðxÞ ¼ rðxÞ;   wðxÞ ¼ r  ðxÞ þ sðxÞ:

ð20Þ

It is well known that the sequence of monic polynomials fqn ðc; xÞgnP0 , orthogonal with respect to dl1 ðc; xÞ, satisfies the structure relation (see [9] and [29])

E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

/ðxÞDðqn ðc; xÞÞ ¼ Aðx; nÞqn ðc; xÞ þ Bðx; nÞqn1 ðc; xÞ;

7119

ð21Þ

where Aðx; nÞ and Bðx; nÞ are polynomials of a fixed degree, and the three term recurrence relation (see [4])

~n q ðc; xÞ þ c ~n qn1 ðc; xÞ; xqn ðc; xÞ ¼ qnþ1 ðc; xÞ þ b n

n P 0;

ð22Þ

with initial conditions q0 ðc; xÞ ¼ 1 and q1 ðc; xÞ ¼ 0, and

pnþ2 ðcÞ pnþ1 ðcÞ ~n ¼ b b  ; nþ1 þ pnþ1 ðcÞ pn ðcÞ

nP0

ð23Þ

and

c~n ¼

pnþ1 ðcÞpn1 ðcÞ ½pn ðcÞ2

cn > 0; n P 1:

ð24Þ

Lemma 1. [17] We have

Aðx; nÞ þ Aðx; n  1Þ þ

~n1 Þ ðx  b Bðx; n  1Þ ¼ /0 ðxÞ  wðxÞ: c~n1

ð25Þ

According to a result by Ismail [17, (1.12)] which must be adapted to our situation since we use monic polynomials, we get

Aðx; nÞ þ Aðx; n  1Þ þ

  ~n1 xb

c~n1

Bðx; n  1Þ ¼ /ðxÞ

½x1 ðxÞ0 wðxÞ  /0 ðxÞ 0 / ðxÞ  wðxÞ;  /ðxÞ /ðxÞ x1 ðxÞ

where x1 ðxÞ ¼ ðx  cÞxðxÞ. 4.1. Uvarov transformations and holonomic equation We consider the Uvarov transformation of the semiclassical measure dlðxÞ ¼ xðxÞ dx. We stablish the holonomic equation of these polynomials. Proof of Theorem 2. Applying the derivative operator in (12) and multiplying it by /ðxÞ, we obtain

/ðxÞDðpn ðk; c; xÞÞ ¼ /ðxÞDðqn ðc; xÞÞ þ cn /ðxÞDðqn1 ðc; xÞÞ:

ð26Þ

Thus, substituting (21) in (26), yields

/ðxÞDðpn ðk; c; xÞÞ ¼ Aðx; nÞqn ðc; xÞ þ ½Bðx; nÞ þ cn Aðx; n  1Þqn1 ðc; xÞ þ cn Bðx; n  1Þqn2 ðc; xÞ: Using (22) in the above expression, we obtain

e nÞq ðc; xÞ þ Bðx; e nÞq ðc; xÞ; /ðxÞðpn ðk; c; xÞÞ0 ¼ Aðx; n n1

ð27Þ

where

e nÞ ¼ Aðx; nÞ  cn Bðx; n  1Þ and Aðx; c~n1  cn  e ~n1 Bðx; n  1Þ: Bðx; nÞ ¼ Bðx; nÞ þ cn Aðx; n  1Þ þ xb c~n1 Therefore, from (12) and (27), it follows that



1

cn

e nÞ Aðx;

e nÞ Bðx;



qn ðc; xÞ qn1 ðc; xÞ



 ¼

pn ðk; c; xÞ /ðxÞDðpn ðk; c; xÞÞ

 ;

that is,

qn ðc; xÞ ¼

e nÞ Bðx; cn /ðxÞ p ðk; c; xÞ  Dðpn ðk; c; xÞÞ e nÞ n e nÞ e nÞ  cn Aðx; e nÞ  cn Aðx; Bðx; Bðx;

and

qn1 ðc; xÞ ¼

e nÞ  Aðx; /ðxÞ p ðk; c; xÞ þ Dðpn ðk; c; xÞÞ: e nÞ n e nÞ e nÞ  cn Aðx; e nÞ  cn Aðx; Bðx; Bðx;

ð28Þ ð29Þ

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E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

Substituting the above two expressions in (21), we deduce

/ðxÞD

e nÞp ðk; c; xÞ Bðx; cn /ðxÞDðpn ðk; c; xÞÞ n  e nÞ Bðx; e nÞ e nÞ  cn Aðx; e nÞ  cn Aðx; Bðx;

!

! e nÞp ðk; c; xÞ Bðx; cn /ðxÞDðpn ðk; c; xÞÞ n  e nÞ Bðx; e nÞ e nÞ  cn Aðx; e nÞ  cn Aðx; Bðx; ! e nÞp ðk; c; xÞ  Aðx; /ðxÞDðpn ðk; c; xÞÞ n þ þ Bðx; nÞ : e nÞ Bðx; e nÞ e nÞ  cn Aðx; e nÞ  cn Aðx; Bðx;

¼ Aðx; nÞ

Then the statement of Theorem 2 follows in a straightforward way. A different approach to this differential equation appears in [26] using the fact that the Uvarov transform of a semiclassical linear functional is again a semiclassical linear functional. If we evaluate the second-order linear differential equation (3) at xn;k ðkÞ :¼ xn;k ðk; cÞ, then we obtain

Aðxn;k ðkÞ; nÞðpn ðk; c; xn;k ðkÞÞÞ00 þ Bðxn;k ðkÞ; nÞðpn ðk; c; xn;k ðkÞÞÞ0 ¼ 0: Hence,

ðpn ðk; c; xn;k ðkÞÞÞ00 Bðxn;k ðkÞ; nÞ ¼ : Aðxn;k ðkÞ; nÞ ðpn ðk; c; xn;k ðkÞÞÞ0

ð30Þ

Substituting Aðxn;k ðkÞ; nÞ and Bðxn;k ðkÞ; nÞ in the right hand side of (30), we get

e n;k ðkÞ; nÞÞ0 Bðx e n;k ðkÞ; nÞ  Bðxn;k ðkÞ; nÞ þ cn Aðxn;k ðkÞ; nÞ  cn /0 ðxn;k ðkÞÞ e n;k ðkÞ; nÞ  cn Aðx ðpn ðk; c; xn;k ðkÞÞÞ00 ð Bðx þ : 0 ¼ e n;k ðkÞ; nÞ e n;k ðkÞ; nÞ  cn Aðx cn /ðxn;k ðkÞÞ ðpn ðk; c; xn;k ðkÞÞÞ Bðx e nÞ and using (25), (28), and (29), then e nÞ  cn Aðx; If we denote Q ðx; nÞ :¼ Bðx;

  cn Qðx; nÞ ¼ Bðx; nÞ þ cn 2Aðx; nÞ þ /0 ðxÞ  wðxÞ þ Bðx; n  1Þ : c~n1

ð31Þ

On the other hand, from (31) and (28), we obtain

e nÞ  Bðx; nÞ þ cn Aðx; nÞ  cn /0 ðxÞ ¼ cn wðxÞ: Bðx; Thus

ðpn ðk; c; xn;k ðkÞÞÞ00 wðxn;k ðkÞÞ ¼ D½ln Q ðx; nÞjx¼xn;k ðkÞ  : /ðxn;k ðkÞÞ ðpn ðk; c; xn;k ðkÞÞÞ0 We consider two external fields



Z

wðxÞ dx and /ðxÞ

ln Q ðx; nÞ:

Thus the total external potential V(x) is given by

VðxÞ ¼ 

Z

wðxÞ dx þ ln Q ðx; nÞ: /ðxÞ

ð32Þ

Let us consider a system of n movable unit charges in (c, b) or (a, c), depending on the location of the point c with respect to (a, b), in the presence of the external potential V(x) given in (32). Let x :¼ ðx1 ; . . . ; xn Þ, where x1 ; . . . ; xn denote the location of the charges. The total energy of the system is

EðxÞ ¼

n X k¼1

Vðxk Þ  2

X

ln jxj  xk j:

16j
In order to find the critical points of E(x) we set



X @ wðxj Þ Q 0 ðxj ; nÞ 1 EðxÞ ¼ 0 () ¼ 0;  þ2 @xj /ðxj Þ Q ðxj ; nÞ x  xk 16k6n;k–j j

Let f ðyÞ :¼ ðy  x1 Þ    ðy  xn Þ. Thus,

wðxj Þ Q 0 ðxj ; nÞ f 00 ðxj Þ  þ ¼ 0; /ðxj Þ Q ðxj ; nÞ f 0 ðxj Þ

j ¼ 1; . . . ; n;

or, equivalently,

f 00 ðyÞ þ

Bðy; nÞ 0 f ðyÞ ¼ 0; Aðy; nÞ

y ¼ x1 ; . . . ; x n :

j ¼ 1; . . . ; n:

ð33Þ

E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

7121

Therefore

f 00 ðyÞ þ

Bðy; nÞ 0 Cðy; nÞ f ðyÞ þ f ðyÞ ¼ 0; Aðy; nÞ Aðy; nÞ

y ¼ x1 ; . . . ; x n :

ð34Þ

On the other hand, from (3) and (34) we obtain f ðyÞ ¼ pn ðk; c; yÞ, which means that the zeros of pn ðk; c; xÞ satisfy (33). h 4.2. Electrostatic interpretation of the zeros of Laguerre type orthogonal polynomials ðaÞ

We give an electrostatic interpretation for the zeros of Laguerre type polynomials pn ðk; c; xÞ which are orthogonal with respect to the measure dlðk; c; xÞ ¼ xa ex dx þ kdðx  cÞ, where c 6 0 and k P 0. We analyze two cases: ðaÞ

1. First, we consider c ¼ 0. Thus, the polynomials pn ðk; 0; xÞ are orthogonal with respect to

dlðk; 0; xÞ ¼ xa ex dx þ kdðxÞ: The measure

dl1 ðxÞ ¼ xaþ1 ex dx; satisfies a Pearson equation with (see (20))

/ðxÞ ¼ rðxÞ ¼ x;

 ðxÞ þ sðxÞ ¼ a þ 2  x: wðxÞ ¼ r

On the other hand, the structure relation (21) reads (see [38]) ðaþ1Þ

/ðxÞDðpðnaþ1Þ ðxÞÞ ¼ Aðx; nÞpnðaþ1Þ ðxÞ þ Bðx; nÞpn1 ðxÞ; where

/ðxÞ ¼ x;

Bðx; nÞ ¼ n þ a þ 1:

Aðx; nÞ ¼ n;

The coefficients (13) and (22) are

c~n ¼ nðn þ a þ 1Þ; ðaÞ

cn ¼ 

1 þ kK n ð0; 0Þ pn1 ð0Þ n!Cða þ 1ÞCða þ 2Þ þ kCðn þ a þ 2Þ nðn þ aÞ ¼ : 1 þ kK n1 ð0; 0Þ pðnaÞ ð0Þ ðn  1Þ!Cða þ 1ÞCða þ 2Þ þ kCðn þ a þ 1Þ

As a conclusion, Q(x, n) in (31) becomes

Q ðx; nÞ ¼ nðn þ a þ 1Þ  cn ð2n þ 1 þ a  cn Þ þ cn x with zero

un ¼ ð2n þ 1 þ a  cn Þ 

nðn þ a þ 1Þ : cn

It is easy to see that 0 < cn < n þ a þ 1. Thus, Q ð0; nÞ < 0 and it implies that un > 0. The electrostatic interpretation of the distribution of zeros means that we have an equilibrium position under the presence of an external potential

ln Q ðx; nÞ  ln xaþ2 ex ; where the first term represents a short range potential corresponding to a unit charge located at un and the second one is a long range potential associated with the weight function (see also [18,19] ). 2. Now, we take c < 0. In this case dl1 ðxÞ ¼ ðx  cÞxa ex dx. Thus, the structure relation (21) for dl1 ðxÞ is

/ðxÞDðqn ðc; xÞÞ ¼ Aðx; nÞqn ðc; xÞ þ Bðx; nÞqn1 ðc; xÞ; where

/ðxÞ ¼ ðx  cÞx;    nþa ; Aðx; nÞ ¼ n x  ðn þ 1 þ an Þ 1 þ an1 nðn þ aÞ Bðx; nÞ ¼ ½an x  ðn þ 1 þ an Þðn þ 1 þ an þ aÞ: an1 From (4) we obtain

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E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127 ðaÞ

ðaÞ

ðx  cÞqn ðc; xÞ ¼ pnþ1 ðxÞ 

pnþ1 ðcÞ ðaÞ

pn ðcÞ

pðnaÞ ðxÞ:

Taking derivatives with respect to x in both hand sides of the above expression, and multiplying the resulting expression by x, we derive ðaÞ

ðaÞ

xqn ðc; xÞ þ xðx  cÞDðqn ðc; xÞÞ ¼ xDðpnþ1 ðxÞÞ 

pnþ1 ðcÞ ðaÞ

pn ðcÞ

xDðpðnaÞ ðxÞÞ:

Using the structure relation and the three therm recurrence relation for Laguerre polynomials we obtain

pnþ1 ðcÞ h ðaÞ

ðaÞ

xqn ðc; xÞ þ xðx  cÞDðqn ðc; xÞÞ ¼ ðn þ 1Þpnþ1 ðxÞ þ ðn þ 1Þðn þ 1 þ aÞpðnaÞ ðxÞ 

ðaÞ pn ðcÞ

i ðaÞ npðnaÞ ðxÞ þ nðn þ aÞpn1 ðxÞ ;

or

"

ðaÞ

xqn ðc; xÞ þ xðx  cÞDqn ðc; xÞ ¼ ðn þ 1Þðx  nÞ 

npnþ1 ðcÞ ðaÞ

pn ðcÞ

# pnðaÞ ðxÞ

 nðn þ 1Þðn þ aÞ þ nðn þ aÞ

! ðaÞ pnþ1 ðcÞ ðaÞ

pn ðcÞ

ðaÞ

pn1 ðxÞ:

Put ða Þ

an ¼

pnþ1 ðcÞ ðaÞ

pn ðcÞ

and, from (15),

pðnaÞ ðxÞ ¼ qn ðc; xÞ 

cn an1

qn1 ðc; xÞ;

we have

xqn ðc; xÞ þ xðx  cÞDðqn ðc; xÞÞ ¼ ððx  nÞðn þ 1Þ  nan Þqn ðc; xÞ  ½ððn þ 1Þðx  nÞ  nan Þ

nðn þ aÞ an1

þ nðn þ aÞðn þ 1 þ an Þqn1 ðc; xÞ þ nðn  1Þðn þ aÞðn  1 þ aÞðn þ 1 þ an Þ

qn2 ðc; xÞ : an2

Using (22) and (23) aÞ ðcÞ c~n1 pnðaÞ ðcÞ pðn2 ; ¼ ðaÞ ð aÞ cn1 pn1 ðcÞ pn1 ðcÞ

we obtain

~n1 an1 c~n1 ðn  1Þðn  1 þ aÞ c ¼ ¼ an2 an2 ðn  1Þðn  1 þ aÞ an1 and, then,

  nðn þ aÞðn þ 1 þ an Þ qn ðc; xÞ xqn ðc; xÞ þ xðx  cÞDðqn ðc; xÞÞ ¼ ððn þ 1Þðx  nÞ  nan Þ  an1   1 ~n1 Þ 1 ððn þ 1Þðx  nÞ  nan Þ  ðn þ 1 þ an Þ q ðc; xÞ: ðn þ 1 þ an Þðx  b þ nðn þ aÞ n1 an1 an1 Therefore,

/ðxÞDðqn ðc; xÞÞ ¼ Aðx; nÞqn ðc; xÞ þ Bðx; nÞqn1 ðc; xÞ; where

/ðxÞ ¼ xðx  cÞ nðn þ aÞðn þ 1 þ an Þ Aðx; nÞ ¼ ððn þ 1Þðx  nÞ  nan Þ   x; an1     1 ~n1  1 ððn þ 1Þðx  nÞ  nan Þ  ðn þ 1 þ an Þ : Bðx; nÞ ¼ nðn þ aÞ ðn þ 1 þ an Þ x  b an1 an1 Simplifying these expressions we derive

E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

7123

   nþa ; Aðx; nÞ ¼ n x  ðn þ 1 þ an Þ 1 þ an1   an n þ 1 þ an ~n1 Þ  ðn þ 1 þ an Þ : xþ ðn  b Bðx; nÞ ¼ nðn þ aÞ an1 an1 In the last expression, using again (23)

~n ¼ b b nþ1 þ anþ1  an ¼ 2n þ a þ 3 þ anþ1  an ; we obtain

Bðx; nÞ ¼

nðn þ aÞ ½an x  ðn þ 1 þ an Þðn þ 1 þ an þ aÞ: an1

This is an alternative approach to the method described in [28]. Notice that the Pearson equation for the linear functional associated with the measure becomes

Dð/u1 Þ ¼ wu1 ; where (see (19))

/ðxÞ ¼ ðx  cÞrðxÞ ¼ ðx  cÞx; wðxÞ ¼ 2rðxÞ þ ðx  cÞsðxÞ ¼ 2x þ ðx  cÞða þ 1  xÞ: According to (23) and (24),

~n1 ¼ b þ an  an1 b n

~n1 ¼ and c

an1 c : an2 n1

This means that Q(x, n) in (31) is the following quadratic polynomial

Q ðx; nÞ ¼ cn x2 þ r n x þ sn with

rn ¼ nðn þ aÞ

an þ c2n  cn ðc þ a þ 1 þ 2nÞ ¼ ðcn þ an Þðcn  an Þ  ðcn  an Þc  ðcn þ an Þð2n þ a þ 1Þ an1

and

sn ¼ ðn þ 1 þ an Þ½ðn þ 1 þ an þ aÞð2n þ 1 þ an þ a  c  2cn Þ þ 2ccn  þ cacn þ c2n ðan  an1 þ 1  cÞ: The zeros of this polynomial are

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r n þ r 2n  4sn cn ; 2cn  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ r n  r 2n  4sn cn : 2cn

z1;n ¼  z2;n

Taking into account

wðxÞ 2 aþ1 ¼ þ  1; /ðxÞ x  c x the electrostatic interpretation means that the equilibrium position for the zeros under the presence of an external potential

ln Q ðx; nÞ  lnðx  cÞ2 xaþ1 ex ; where the first one is a short range potential corresponding to two unit charges located at z1;n and z2;n and the second one is a long range potential associated with a polynomial perturbation of the weight function. 4.3. Electrostatic interpretation for the zeros of Jacobi type orthogonal polynomials ða;bÞ

We furnish an electrostatic interpretation for the zeros of Jacobi type polynomials pn respect to the measure

ðk; c; xÞ which are orthogonal with

dlðk; c; xÞ ¼ ð1  xÞa ð1 þ xÞb dx þ kdðx  cÞ with c R ð1; 1Þ and k P 0. For this propose we separate in two cases: ða;bÞ

1. First, we consider c ¼ 1. Thus, the polynomials pn

ðk; 1; xÞ are orthogonal with respect to

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E.J. Huertas et al. / Applied Mathematics and Computation 218 (2012) 7109–7127

dlðk; 1; xÞ ¼ ð1  xÞa ð1 þ xÞb dx þ kdðx þ 1Þ: The measure

dl1 ðxÞ ¼ ðx  ð1ÞÞ dlðxÞ ¼ ð1  xÞa ð1 þ xÞbþ1 dx; satisfies a Pearson equation with (see (20))

/ðxÞ ¼ rðxÞ ¼ 1  x2 ;

 ðxÞ þ sðxÞ ¼ ðb  a þ 1Þ  ða þ b þ 3Þx: wðxÞ ¼ r

On the other hand, the structure relation (21) reads ða;bþ1Þ

/ðxÞDðpðna;bþ1Þ ðxÞÞ ¼ Aðx; nÞpnða;bþ1Þ ðxÞ þ Bðx; nÞpn1

ðxÞ;

where

Aðx; nÞ ¼ Bðx; nÞ ¼

n½b  a þ 1 þ ð2n þ a þ b þ 1Þx ; 2n þ a þ b þ 1 4nðn þ aÞðn þ b þ 1Þðn þ a þ b þ 1Þ ð2n þ a þ b þ 1Þ2 ð2n þ a þ bÞ

:

~n in (22) when qn ð1; xÞ ¼ pnða;bþ1Þ ðxÞ is The coefficient c

c~n ¼ can;bþ1 ¼

4nðn þ aÞðn þ b þ 1Þðn þ a þ b þ 1Þ ð2n þ a þ bÞð2n þ a þ b þ 1Þ2 ð2n þ a þ b þ 2Þ

and

cn ¼

1 þ kK n ð1; 1Þ 2nðn þ aÞ > 0; 1 þ kK n1 ð1; 1Þ ð2n þ a þ bÞð2n þ a þ b þ 1Þ

Thus,

Qðx; nÞ ¼ Bðx; nÞ þ cn ½ð2n þ a þ bÞcn 

ða þ b þ 1Þðb  a þ 1Þ  þ ð2n þ a þ b þ 1Þcn x: 2n þ a þ b þ 1

Observe that the zero of Q(x, n) belongs to ð1; 1Þ. In fact, after some tedious calculations we see that

Qð1Þ ¼ Bð1; nÞ þ cn ½ð2n þ a þ bÞcn þ

2ð2nðn þ a þ b þ 1Þ þ aða þ b þ 1ÞÞ >0 2n þ a þ b þ 1

and

Qð1Þ ¼

2aþbþ3 ðb þ 1ÞCðnÞCðn þ aÞCðb þ 2Þ2 Cðn þ b þ 1ÞCðn þ a þ b þ 1Þk 2n þ a þ b 1 < 0:  aþbþ1 2 CðnÞCðn þ aÞCðb þ 1ÞCðb þ 2Þ þ kCðn þ b þ 1ÞCðn þ a þ b þ 1Þ

Using some known properties of the Jacobi polynomials we conclude that ða;bÞ

nðn þ aÞðpn

un ¼ 1 þ 2k



ða;bÞ 2 ð1ÞÞ2 = pn

2

ða;bÞ

ðpn



ða;bÞ 2 ð1ÞÞ2 = pn

3

6 7 l 7 6 4K n1 ð1; 1Þð1 þ kK n1 ð1; 1ÞÞ5: ð2n þ a þ b þ 1Þ ð1 þ kK n ð1; 1ÞÞ l

2

The electrostatic interpretation means that the equilibrium position for the zeros under the presence of an external potential

ln Q ðx; nÞ  lnð1  xÞaþ1 ð1 þ xÞbþ2 ; where the first one is a short range potential corresponding to a unit charge located at the zero of Q(x,n) and the other one is a long range potential associated with the weight function. 2. We take c < 1. Then,

dl1 ðxÞ ¼ ðx  cÞð1  xÞa ð1 þ xÞb dx and the structure relation (21) for the above measure is

/ðxÞDðqn ðc; xÞÞ ¼ Aðx; nÞqn ðc; xÞ þ Bðx; nÞqn1 ðc; xÞ; where

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/ðxÞ ¼ ðx  cÞð1  x2 Þ; Aðx; nÞ ¼ anþ1 ðx  bn Þ þ bnþ1  kn an  ðanþ1 cn þ kn bn Þ Bðx; nÞ ¼ ðanþ1 ðx  bn Þ þ bnþ1  kn an Þ

cn kn1

1  1 þ x2 ; kn1

þ anþ1 cn þ kn bn  ðanþ1 cn þ kn bn Þ

~n1 xb : kn1

Put ða;bÞ

kn ¼ kna;b ðcÞ ¼

pnþ1 ðcÞ ða;bÞ

pn

ðcÞ

ð35Þ

:

From (4) we obtain ða;bÞ

ðx  cÞqn ðc; xÞ ¼ pnþ1 ðxÞ  kn pðna;bÞ ðxÞ: Taking derivatives with respect to x in both hand sides of the above expression, and multiplying them by 1  x2 , we see that ða;bÞ

ð1  x2 Þqn ðc; xÞ þ ðx  cÞð1  x2 ÞDðqn ðc; xÞÞ ¼ ð1  x2 ÞDðpnþ1 ðxÞÞ  kn ð1  x2 ÞDðpðna;bÞ ðxÞÞ: Since ða;bÞ

ð1  x2 ÞDðpðna;bÞ ðxÞÞ ¼ an pðna;bÞ ðxÞ þ bn pn1 ðxÞ; where

an ¼ ana;b ðxÞ ¼

4nðn þ aÞðn þ bÞðn þ a þ bÞ

a;b

bn ¼ bn ¼

n½b  a þ ð2n þ a þ bÞx ; 2n þ a þ b

ð2n þ a þ bÞ2 ð2n þ a þ b  1Þ

;

we obtain ða;bÞ

ða;bÞ

ð1  x2 Þqn ðc; xÞ þ ðx  cÞð1  x2 ÞDðqn ðc; xÞÞ ¼ anþ1 pnþ1 ðxÞ þ ðbnþ1  kn an Þpðna;bÞ ðxÞ  kn bn pn1 ðxÞ: The three terms recurrence relation of monic Jacobi polynomials implies ða;bÞ

ð1  x2 Þqn ðc; xÞ þ ðx  cÞð1  x2 ÞDðqn ðc; xÞÞ ¼ ½anþ1 ðx  bn Þ þ bnþ1  kn an pðna;bÞ ðxÞ  ðanþ1 cn þ kn bn Þpn1 ðxÞ: From (15) and (35),

pðna;bÞ ðxÞ ¼ qn ðc; xÞ 

cn kn1

qn1 ðc; xÞ:

Then

ð1  x2 Þqn ðc; xÞ þ ðx  cÞð1  x2 ÞDðqn ðc; xÞÞ ¼ ½anþ1 ðx  bn Þ þ bnþ1  kn an qn ðc; xÞ   c  ðanþ1 ðx  bn Þ þ bnþ1  kn an Þ n þ anþ1 cn þ kn bn qn1 kn1  ðc; xÞðanþ1 cn þ kn bn Þ

cn1 kn2

qn2 ðc; xÞ:

From (22) for monic kernels,

ð1  x2 Þqn ðc; xÞ þ ðx  cÞð1  x2 ÞDðqn ðc; xÞÞ ¼ ½anþ1 ðx  bn Þ þ bnþ1  kn an qn ðc; xÞ   c  ðanþ1 ðx  bn Þ þ bnþ1  kn an Þ n þ anþ1 cn þ kn bn qn1 ðc; xÞ kn1   ~ cn1 x  bn1 qn1 ðc; xÞ  qn ðc; xÞ :  ðanþ1 cn þ kn bn Þ kn2 c~n1 According to (23) and (35), we obtain

c~n1 kn1 ¼ : cn1 kn2 Therefore

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  1 q ðc; xÞ ð1  x2 Þqn ðc; xÞ þ ðx  cÞð1  x2 ÞDðqn ðc; xÞÞ ¼ anþ1 ðx  bn Þ þ bnþ1  kn an  ðanþ1 cn þ kn bn Þ kn1 n "  ðanþ1 ðx  bn Þ þ bnþ1  kn an Þ

cn

kn1

þ anþ1 cn þ kn bn  ðanþ1 cn þ kn bn Þ

~n1 xb kn1

#

 qn1 ðc; xÞ: Thus

/ðxÞDðqn ðc; xÞÞ ¼ Aðx; nÞqn ðc; xÞ þ Bðx; nÞqn1 ðc; xÞ; where

/ðxÞ ¼ ðx  cÞð1  x2 Þ; Aðx; nÞ ¼ anþ1 ðx  bn Þ þ bnþ1  kn an  ðanþ1 cn þ kn bn Þ Bðx; nÞ ¼ ðanþ1 ðx  bn Þ þ bnþ1  kn an Þ

cn kn1

1  1 þ x2 ; kn1

þ anþ1 cn þ kn bn  ðanþ1 cn þ kn bn Þ

~n1 xb : kn1

Simplifying these expressions we have

Aðx; nÞ ¼ An;0 þ An;1 x þ An;2 x2 and

Bðx; nÞ ¼ Bn;0 þ Bn;1 x þ Bn;2 x2 : Notice that the Pearson equation for the linear functional associated with the measure

dl1 ðxÞ ¼ ðx  cÞð1  xÞa ð1 þ xÞb dx; becomes

Dð/u1 Þ ¼ wu1 ; with (see (19))

/ðxÞ ¼ ðx  cÞrðxÞ ¼ ðx  cÞð1  x2 Þ;

wðxÞ ¼ 2rðxÞ þ ðx  cÞsðxÞ ¼ 2ð1  x2 Þ þ ðx  cÞðb  a  ða þ b þ 2ÞxÞ;

which means that Q(x, n) is the following quadratic polynomial

  cn Qðx; nÞ ¼ Bðx; nÞ þ cn 2Aðx; nÞ þ /0 ðxÞ  wðxÞ þ Bðx; n  1Þ c~n1      2 cn Bn1;2 c Bn1;1 þ Bn;1  ½aðc  1Þ þ bðc þ 1Þ þ 2An;1 cn x cn x2 þ n ¼ Bn;2 þ a þ b þ 1  2An;2 þ c~n1 c~n1 þ Bn;0  ½2An;0 þ 1 þ cða  bÞcn þ

c2n Bn1;0 : c~n1

Taking into account

wðxÞ 2 aþ1 bþ1 ¼  þ ; /ðxÞ x  c 1  x 1 þ x the electrostatic interpretation means that the equilibrium position for the zeros under the presence of an external potential

ln Q ðx; nÞ  lnðx  cÞ2 ð1  xÞaþ1 ð1 þ xÞbþ1 ; where the first one is a short range potential corresponding to two unit charges located at the zeros of Q(x, n) and the second one is a long range potential associated with a polynomial perturbation of the weight function. References [1] R. Álvarez-Nodarse, F. Marcellán, J. Petronilho, WKB approximation and Krall-type orthogonal polynomials, Acta Appl. Math. 54 (1998) 27–58. [2] C. Bernardi, Y. Maday, Approximations Spectrales de Problèmes aux Limites Elliptiques, Mathematics and Applications, vol. 10, Springer-Verlag, Paris, 1992 (in French). [3] C.F. Bracciali, D.K. Dimitrov, A. Sri Ranga, Chain sequences and symmetric generalized orthogonal polynomials, J. Comput. Appl. Math. 143 (2002) 95– 106. [4] M.I. Bueno, F. Marcellán, Darboux transformations and perturbation of linear functionals, Lin. Alg. Appl. 384 (2004) 215–242. [5] T.S. Chihara, An Introduction to Orthogonal Polynomials. Mathematics and its Applications Series, Gordon and Breach, New York, 1978. [6] D.K. Dimitrov, F. Marcellán, F.R. Rafaeli, Monotonicity of zeros of Laguerre–Sobolev type orthogonal polynomials, J. Math. Anal. Appl. 368 (2010) 80–89. [7] D.K. Dimitrov, M.V. Mello, F.R. Rafaeli, Monotonicity of zeros of Jacobi–Sobolev type orthogonal polynomials, Appl. Numer. Math. 60 (2010) 263–276. [8] D.K. Dimitrov, W. Van Assche, Lamé differential equations and electrostatics, Proc. Amer. Math. Soc. 128 (2000) 3621–3628.

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