Asymptotic behavior of varying discrete Jacobi-Sobolev orthogonal polynomials

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently. (C) 2016 Elsevier B.V. All rights reserved.The authors JFMM and JJMB are partially supported by Research Group FQM-0229 (belonging to Campus of International Excellence CEIMAR). The author JFMM is funded by a grant of Plan Propio de la Universidad de Almería. The author FM is partially supported by Dirección General de Investigación, Ministerio de Economía y Competitividad Innovación of Spain, Grant MTM2012-36732-C03-01. The author JJMB is partially supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain and European Regional Development Found, grants MTM2011-28952-C02-01 and MTM2014-53963-P, and Junta de Andalucía (excellence grant P11-FQM-7276)

Varying discrete Laguerre-Sobolev orthogonal polynomials: Asymptotic behavior and zeros

We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler-Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give us an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other special functions. We generalize some results appeared very recently in the literature for both the varying and non-varying cases.The author FM is partially supported by Dirección General de Investigación, Ministerio de Economía y Competitividad Innovación of Spain, Grant MTM2012 36732 C03 01. The author JJMB is partially supported by Dirección General de Inves tigación, Ministerio de Ciencia e Innovación of Spain and European Regional Development Found, Grant MTM2011 28952 C02 01, and Junta de Andalucía, Research Group FQM 0229 (belonging to Campus of International Excellence CEI MAR), and projects P09 FQM 4643 and P11 FQM 7276

A differential equation for varying Krall type orthogonal polynomials

In this contribution we consider varying Krall--type polynomials which are orthogonal with respect to a varying discrete Krall--type inner product. Our main goal is to give ladder operators for this family of polynomials as well as to find a second--order differential--difference equation that these polynomials satisfy. We generalize some results appeared recently in the literature

Asymptotics for Some q-Hypergeometric Polynomials

We tackle the study of a type of local asymptotics, known as Mehler–Heine asymptotics, for some q–hypergeometric polynomials. Some consequences about the asymptotic behavior of the zeros of these polynomials are discussed. We illustrate the results with numerical examples

Asymptotics for varying discrete Sobolev orthogonal polynomials

We consider a varying discrete Sobolev inner product such as $$(f,g)_S=\int f(x)g(x)d \mu+M_nf^{(j)}(c)g^{(j)}(c),$$ where $\mu$ is a finite positive Borel measure supported on an infinite subset of the real line, $c$ is adequately located on the real axis, $j \geq0,$ and $\{M_n\}_{n\geq0}$ is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler--Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around $c,$ just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler--Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones

Ladder operators and a differential equation for varying generalized Freud type orthogonal polynomials

In this paper we introduce varying generalized Freud--type polynomials which are orthogonal with respect to a varying discrete Freud--type inner product. Our main goal is to give ladder operators for this family of polynomials as well as finding a second order differential--difference equation that these polynomials satisfy. To reach this objective it is necessary to consider the standard Freud orthogonal polynomials and in the meanwhile we find new difference relations for the coefficients in the first order differential equations that this standard family satisfies

Characterization of orthogonal polynomials on lattices

We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ such that \begin{align*} \sum_{j=1} ^{M} a_{j,n}\mathrm{S}_x\mathrm{D}_x ^k P_{k+n-j} (z)=\sum_{j=1} ^{N} b_{j,n}\mathrm{D}_x ^{m} Q_{m+n-j} (z)\;, \end{align*} with $k,m,M,N \in \mathbb{N}$, $a_{j,n}$ and $b_{j,n}$ are sequences of complex numbers, $$2\mathrm{S}_xf(x(s))=(\triangle +2\mathrm{I})f(z),\quad \mathrm{D}_xf(x(s))=\frac{\triangle}{\triangle x(s-1/2)}f(z),$$ $z=x(s-1/2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $\triangle f(s)=f(s+1)-f(s)$. We show that under some natural conditions, both involved orthogonal polynomials sequences $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ are semiclassical whenever $k=m$. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices

Second-Order Difference Equation for Sobolev-Type Orthogonal Polynomials. Part II: Computational Tools

We consider polynomials which are orthogonal with respect to a nonstandard inner product. In fact, we deal with Sobolev-type orthogonal polynomials in the broad sense of the expression. This means that the inner product under consideration involves the Hahn difference operator, thus including the difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. In a previous work, we studied properties of these polynomials from a theoretical point of view. There, we obtained a second-order differential/difference equation satisfied by these polynomials. The aim of this paper is to present an algorithm and a symbolic computer program that provides us with the coefficients of the second-order differential/difference equation in this general context. To illustrate both, the algorithm and the program, we will show three examples related to different operators