A kind of orthogonal polynomials and related identities II

TL;DR

This paper explores the properties of a specific family of orthogonal polynomials, establishing their connection to Meixner polynomials, deriving new formulas and recurrence relations, and analyzing their positivity properties.

Contribution

It introduces new formulas and recurrence relations for the polynomials $d_n^{(r)}(x)$ and provides a novel proof for their squared form, expanding understanding of their structure and properties.

Findings

Connection established between $d_n^{(r)}(x)$ and Meixner polynomials

New formulas and recurrence relations derived for $d_n^{(r)}(x)$

Positivity properties of $d_n^{(r)}(x)$ established for certain ranges of $x$

Abstract

For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given. In addition, for $r>-\frac 12$ and $n\ge 2$ we show that $d_n^{(r)}(x)>\frac{(2x+1)^n}{n!}>0$ for $x>-\frac 12$, and $(-1)^nd_n^{(r)}(x)>0$ for $x<-\frac 12$.