A kind of orthogonal polynomials and related identities II

Abstract

For $n=0,1,2,\dots$ let $d_n^{(r)}(x)={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{x+r+k}{k}\right)\left(\genfrac{}{}{0ex}{}{x-r}{n-k}\right)$. 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 $r>-\frac{1}{2}$ and $n\ge 2$ we show that $d_n^{(r)}(x)>\frac{(2x+1{)}^n}{n!}>0$ for $x>-\frac{1}{2}$, and $(-1{)}^n{d}_n^{(r)}(x)>0$ for $x<-\frac{1}{2}$.

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