A kind of orthogonal polynomials and related identities II

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 $x<-\frac 12$.