New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

Abstract

In this contribution, we consider the sequence $\{Q_{n}^{\lambda }\}_{n\geq 0}$ of monic polynomials orthogonal with respect to the following inner product involving differences

$$\langle p,q\rangle_{\lambda }={\int}_{0}^{\infty }p\left( x\right) q\left( x\right) d\psi^{(a)}(x)+\lambda {\Delta} p(c){\Delta} q(c), $$

where $\lambda \in \mathbb {R}_{+}$, Δ denotes the forward difference operator defined by Δf (x) = f (x + 1) − f (x), ψ^(a) with a > 0 is the well-known Poisson distribution of probability theory

$$d\psi^{(a)}(x)=\frac{e^{-a}a^{x}}{x!}\quad \text{at }x = 0,1,2,{\ldots} , $$

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.