On inversion and connection coefficients for basic hypergeometric polynomials

In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric polynomials and we derive expansion formulae corresponding to all the families within the $q$-Askey scheme.Comment: 15 page

Classification of 2-Orthogonal Polynomials with Brenke Type Generating Functions

The Brenke type generating functions are the polynomial generating functions of the form $$\sum_{n=0}^{\infty}{P_n(x )\over n!}t^n=A(t)B(xt),$$ where $A$ and $B$ are two formal power series subject to the conditions $A(0)\;B^{(k)}(0)\neq0,\, k=0,1,2\ldots$.\\ In this work, we determine all Brenke-type polynomials when they are also $2$-orthogonal polynomial sets, that is to say, polynomials satisfying one standard four-term recurrence relation. That allows us, on one hand, to obtain new 2-orthogonal sequences generalizing known orthogonal families of polynomials, and on the other hand, to recover particular cases of polynomial sequences discovered in the context of $d$-orthogonality.\\ The classification is based on the resolution of a three-order difference equation induced by the four-term recurrence relation satisfied by the considered polynomials. This study is motivated by the work of Chihara who gave all pairs $(A (t), B(t))$ for which $\{P_n(x)\}_{n\geq 0}$ is an orthogonal polynomial sequence.Comment: 22 page

Operational rules and a generalized Hermite polynomials

AbstractIn this paper, we use operational rules associated with three operators corresponding to a generalized Hermite polynomials introduced by Szegö to derive, as far as we know, new proofs of some known properties as well as new expansions formulae related to these polynomials

Duplication Coefficients via Generating Functions

... Xn Cmðn, aÞPmðxÞ, m0 where fPngn 0 belongs to a wide class of polynomials, including the classical orthogonal polynomials (Hermite, Laguerre, Jacobi) as well as the classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouk) for the specific case a 1. We give closed-form expressions as well as recurrence relations satisfied by the duplication coefficients

On Linearization Coefficients of Jacobi Polynomials

This article deals with the problem of finding closed analytical formulae for generalized linearization coefficients for Jacobi polynomials. By considering some special cases we obtain a reduction formula using for this purpose symbolic computation, in particular Zeilberger’s and Petkovsek’s algorithms

On linearization and connection coefficients for generalized Hermite polynomials

AbstractWe consider the problem of finding explicit formulas, recurrence relations and sign properties for both connection and linearization coefficients for generalized Hermite polynomials. Most of the computations are carried out by the computer algebra system Maple using appropriate algorithms