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.

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Acknowledgments

We would like to thank the anonymous referees for carefully reading the manuscript and for giving constructive comments, which substantially helped us to improve the quality of the paper. The work of the first author was partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grant MTM2015-65888-C4-2-P.

Funding

The work of the first author was partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grant MTM2015-65888-C4-2-P.

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Correspondence to Edmundo J. Huertas.

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Huertas, E.J., Soria-Lorente, A. New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials. Numer Algor 82, 41–68 (2019). https://doi.org/10.1007/s11075-018-0593-0

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Keywords

  • Charlier polynomials
  • Sobolev-type polynomials
  • Discrete kernel polynomials
  • Discrete quasi-orthogonal polynomials

Mathematics Subject Classification (2010)

  • 33C47