A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

Abstract

In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the Δ-Meixner-Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre-Sobolev orthogonal polynomials.

New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

“…n−2 (x). Combining all the above expressions into (21) we obtain the desired coefficients in (24). To obtain (25), it is enough to consider Proposition 2.…”

“…Since then, and to the best of our knowledge, the Charlier case has remained untouched. Several researchers have done further work on the Sobolev-type case for discrete orthogonality measures, but mainly concerning the Meixner case (see [4], [5], [15], [20], [21] and the references given there).…”

Mehler–Heine type formulas for Charlier and Meixner polynomials

“…The work presented in this paper was motivated by a question that Professor Juan José Moreno-Balcázar asked during our visit to the Universidad de Almería in 2010. He was wondering if it would be possible to have Mehler-Heine type formulas for discrete orthogonal polynomials, since this would help in calculations involving asymptotics of discrete Sobolev polynomials [45].…”

Uniform Asymptotic Approximations for the Meixner–sobolev Polynomials

“…When λ = 0 the Meixner-Sobolev polynomials reduce to the classical Meixner polynomials. These polynomials were introduced in [1], and a recurrence relation involving S n , S n−1 and 2 classical Meixner polynomials are given in [2] and [7]. This recurrence relation is very useful for generating the polynomials.…”

“…In this paper we give large n asymptotic approximations that are valid on the real x axis, hence, they include the oscillatory region 0 x/n < 1+ √ c 1− √ c . The starting point is the generating function given in [7]:…”