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A kind of orthogonal polynomials and related identities (1606.08327v3)

Published 27 Jun 2016 in math.NT, math.CA, and math.CO

Abstract: In this paper we introduce the polynomials ${d_n{(r)}(x)}$ and ${D_n{(r)}(x)}$ given by $d_n{(r)}(x)=\sum_{k=0}n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0{(r)}(x)=1,\ D_1{(r)}(x)=x$ and $D_{n+1}{(r)}(x)=xD_n{(r)}(x)-n(n+2r)D_{n-1}{(r)}(x)\ (n\ge 1).$ We show that ${D_n{(r)}(x)}$ are orthogonal polynomials for $r>-\frac 12$, and establish many identities for ${d_n{(r)}(x)}$ and ${D_n{(r)}(x)}$, especially obtain a formula for $d_n{(r)}(x)2$ and the linearization formulas for $d_m{(r)}(x)d_n{(r)}(x)$ and $D_m{(r)}(x)D_n{(r)}(x)$. As an application we extend recent work of Sun and Guo.

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