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Suppose $$\{P_{n}^{(\alpha , \beta )}(x)\} _{n=0}^\infty $$ is a sequence of Jacobi polynomials with $$ \alpha , \beta >-1.$$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+k}^{(\alpha + t, \beta + s )}(x)$$ are interlacing if $$s,t >0$$ and $$ k \in {\mathbb {N}}.$$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+1}^{(\alpha , \beta + 1 )}(x),$$ $$ \alpha> -1, \beta > 0, $$ $$ n \in {\mathbb {N}},$$ are partially, but in general not fully, interlacing depending on the values of $$\alpha , \beta $$ and n. A similar result holds for the extent to which interlacing holds between the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+1}^{(\alpha + 1, \beta + 1 )}(x),$$ $$ \alpha>-1, \beta > -1.$$ It is known that the zeros of the equal degree Jacobi polynomials $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n}^{(\alpha - t, \beta + s )}(x)$$ are interlacing for $$ \alpha -t> -1, \beta > -1, $$ $$0 \le t,s \le 2.$$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n}^{(\alpha + 1, \beta + 1 )}(x),$$ when $$ \alpha> -1, \beta > -1.$$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $$\alpha = \beta = \lambda -1/2$$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $$ C_{n}^{(\lambda )}(x)$$ and $$ C_{n + 1}^{(\lambda +1)}(x),$$ $$ \lambda > -1/2,$$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $$ C_{n}^{(\lambda )}(x)$$ and $$ C_{n}^{(\lambda +3)}(x),$$ $$ \lambda > -1/2,$$ is also discussed.

We investigate interlacing properties of zeros of Laguerre polynomials $ L_{n}^{(\alpha)}(x)$ and $ L_{n+1}^{(\alpha +k)}(x),$ $ \alpha > -1, $ where $ n \in \mathbb{N}$ and $ k \in {\{ 1,2 }\}$. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp $t-$interval within which the zeros of two equal degree Laguerre polynomials $ L_n^{(\alpha)}(x)$ and $ L_n^{(\alpha +t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ \alpha > -1$ is $ 0 < t \leq 2,$ \cite{DrMu2}, and the sharp $t-$interval within which the zeros of two consecutive degree Laguerre polynomials $ L_n^{(\alpha)}(x)$ and $ L_{n-1}^{(\alpha +t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ \alpha > -1$ is $ 0 \leq t \leq 2,$ \cite{DrMu1}. We derive conditions on $n \in \mathbb{N}$ and $\alpha,$ $ \alpha > -1$ that determine the partial or full interlacing of the zeros of $ L_n^{(\alpha)}(x)$ and the zeros of $ L_n^{(\alpha + 2 + k)}(x),$ $ k \in {\{ 1,2 }\}$. We also prove that partial interlacing holds between the zeros of $ L_n^{(\alpha)}(x)$ and $ L_{n-1}^{(\alpha + 2 +k )}(x)$ when $ k \in {\{ 1,2 }\},$ $n \in \mathbb{N}$ and $ \alpha > -1$. Numerical illustrations of interlacing and its breakdown are provided.

We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribu...

We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained.

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to &zeta, ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

In this paper, we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e. this function solves a first-order non-homogeneous q-difference equation. The solutions of the aforementioned q-difference equation (given in terms of hypergeometric series) for some canonical cases, namely, q-Charlier, q-Kravchuk, q-Meixner and q-Hahn, are worked out.

We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula. The research of J. Arves u was partially supported by the research grant MTM2012-36732-C03-01 (Ministerio de Econom a y Competitividad) of Spain.

20 pages, no figures.-- MSC2000 codes: 33C45, 33A65, 42C05. MR#: MR2061464 (2005b:33007) Zbl#: Zbl 1089.33005 We consider the general theory of the modifications of quasi-definite linear functionals by adding discrete measures. We analyze the existence of the corresponding orthogonal polynomial sequences with respect to such linear functionals. The three-term recurrence relation, lowering and raising operators as well as the second order linear differential equation that the sequences of monic orthogonal polynomials satisfy when the linear functional is semiclassical are also established. A relevant example is considered in details. This work is partially supported by Dirección General de Investigacion (Ministerio de Ciencia y Tecnología) of Spain BFM 2000-0206-C04, Junta de Andalucía FQM-0262, and INTAS n° 2000-272. Publicado

27 pages, no figures.-- MSC2000 codes: 33C45, 33C10, 42C05, 41A28.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001). MR#: MR1985676 (2004g:33015) Zbl#: Zbl 1021.33006 In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317–347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2. This research was supported by INTAS project 00-272, Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM-2000-0029 and BFM-2000-0206-C04-01, Dirección General de Investigación de la Comunidad Autónoma de Madrid, and by project G.0184.02 of FWO-Vlaanderen. Publicado

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional UU=Jα,β+A1δ(x−1)+B1δ(x+1)−A2δ′(x−1)−B2δ′(x+1), where Jα,β is the Jacobi linear functional, i.e. 《Jα,β,p›=∫−11p(x)(1−x)α(1+x)β dx,αα,β>−1, p∈P, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1,1) (inner asymptotics) and C∖[−1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A2=B2=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n−1/n] Pade approximants are our orthogonal polynomials.

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L2(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L2(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A1, first observed by Everitt, and the self-adjoint operator A1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Maric result which gives optimal global smoothness of functions in D(A).

We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arves\'u et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the equilibrium problem with constrain to describe their zero distribution. Then analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics., Comment: The n-root asymptotic behavior of multiple Meixner polynomials is studied. A method based on an algebraic function formulation in connection with some available techniques from logarithmic potential theory has been developed. It represents an alternative to the use of Riemann-Hilbert techniques and the steepest descent method for oscillatory RH problems, when dealing with multiple orthogonality

In the present paper, starting from the second-order difference hypergeometric equation on the non-uniform lattice x(s) satisfied by the set of discrete hypergeometric orthogonal q-polynomials {pn}, we find analytical expressions of the expansion coefficients of any q-polynomial rm(x(s)) on x(s) and of the product rm(x(s))qj(x(s)) in series of the set {pn}. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials. Unión Europea Dirección General de Enseñanza Superior Junta de Andalucía

11 pages, no figures.-- MSC2000 codes: 33C45, 42C05, 33C47, 47A06. MR#: MR2108368 (2005m:33012) Zbl#: Zbl 1080.33008 Using a general and simple algebraic approach, some results on Krall-type orthogonal polynomials and some of their extensions are obtained. This work was partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain BFM 2003-06335-C03, Junta de Andalucía FQM-0262 (RAN), and INTAS Research Network NeCCA (INTAS 03-51-6637) (FM and JA). Publicado

26 pages, no figures.-- MSC1991 code: 33D25. MR#: MR1771452 (2001b:33022) Zbl#: Zbl 0956.33009 ^aThe main goal of this paper is to continue the study of q-polynomials on non-uniform lattices by using the approach introduced by A. F. Nikiforov and V. B. Uvarov [Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1983, no. 17, 34 pp.; MR0753537 (86c:39006)]. We consider the q-polynomials on the non-uniform exponential lattice $x(s)=c_1q^s+c_3$ and study some of their properties (differentiation formulas, structure relations, representation in terms of hypergeometric and basic hypergeometric functions, etc.). Special emphasis is given to q-analogues of the Charlier orthogonal polynomials. For these Charlier polynomials we compute the main data, i.e., the coefficients of the three-term recurrence relation, the structure relation, the square of the norm, etc., in the exponential lattices $x(s)=q^s$ and $x(s)=(q^s-1)/(q-1)$, respectively. This work was completed while one of the authors (RAN) was visiting the Universidade de Coimbra. He is very grateful to the Department of Mathematics of Universidade de Coimbra for the kind hospitality and the Centro de Matematica da Universidade de Coimbra for financial support. The research of the authors was partially supported by Dirección General de Enseñanza Superior (DGES) PB 96-0120-C03-01 and the European project INTAS 93-219-ext. Publicado

24 pages, no figures.-- MSC1991 codes: 33C45, 33A65, 42C05. MR#: MR1775827 (2001b:33013) Zbl#: Zbl 0936.33003 We consider the perturbation of the classical Bessel moment functional by the addition of the linear functional $M_0\delta(x)+M_1\delta'(x)$, where $M_0$ and $M_1\in\bf R$. We give necessary and sufficient conditions in order for this functional to be a quasi-definite functional. In such a situation we analyze the corresponding sequence of monic orthogonal polynomials $B^{\alpha,M_0,M_1}_n(x)$. In particular, a hypergeometric representation $(_4F_2)$ for them is obtained. Furthermore, we deduce a relation between the corresponding Jacobi matrices, as well as the asymptotic behavior of the ratio $B^{\alpha,M_0,M_1}_n(x)/B^\alpha_n(x)$, outside the closed contour $\Gamma$ containing the origin, and the difference between the new polynomials and the classical ones, inside $\Gamma$. The work of the first three authors was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB 96-0120-C03-01. The fourth author (KHK) thanks KOSEF(95-070-02-01-3) and Korea Ministry of Education (BSR1 1420) for their research support. Publicado

22 pages, 4 figures.-- MSC1991 codes: 33C45; 33A65; 42C05.-- Dedicated to Professor Mario Rosario Occorsio on his 65th birthday. MR#: MR1624329 (99h:33029) Zbl#: Zbl 0924.33006 We obtain an explicit expression for the Sobolev-type orthogonal polynomials $\{Q_n\}$ associated with the inner product $\langle p,q\rangle=\int^1_{-1}p(x)q(x)\rho(x)dx+A_1p(1)q(1)+B_1p(-1)q(-1)+A_2p'(1)q'(1)+B_2p'(-1)q'(-1)$, where $\rho(x)=(1-x)^\alpha(1+x)^\beta$ is the Jacobi weight function, $\alpha,\beta>-1$, $A_1,B_1,A_2,B_2\geq 0$ and $p,q\in\bold P$, the linear space of polynomials with real coefficients. The hypergeometric representation $({}_6F_5)$ and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in $[-1,1]$ is studied. Furthermore, we obtain some estimates for the largest zero of $Q_n(x)$. Such a zero is located outside the interval $[-1,1]$. We deduce its dependence on the masses. Finally, the WKB analysis for the distribution of zeros is presented. The research of the first author (J.A.) was supported by a grant of Ministerio de Educación y Cultura (MEC) of Spain. The research of the three first authors (J.A., R.A.N. and F.M.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB 96-0120-C03-01 and INTAS Project INTAS 93-0219 Ext. Publicado