New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

Abstract

In this contribution we consider the sequence $\{Q_{n}^λ\}_{n\geq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences \begin{equation*} \langle p,q\rangle _λ=\int_{0}^{\infty}p\left(x\right) q\left(x\right) dψ^{(a)}(x)+λ\,Δp(c)Δq(c), \end{equation*} where $λ\in \mathbb{R}_{+}$, $Δ$ denotes the forward difference operator defined by $Δf\left(x\right) =f\left(x+1\right) -f\left(x\right) $, $ψ^{(a)}$ with $a>0$ is the well known Poisson distribution of probability theory% \begin{equation*} dψ^{(a)}(x)=\frac{e^{-a}a^{x}}{x!}\quad \text{at}x=0,1,2,\ldots, \end{equation*}% and $c\in \mathbb{R}$ is such that $ψ^{(a)}$ has no points of increase in the interval $(c,c+1)$. We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of $c$ such that $c+1<0$, we obtain some results on the distribution of its zeros as decreasing functions of $λ$, when this parameter goes from zero to infinity.