A kind of orthogonal polynomials and related identities

TL;DR

This paper introduces new orthogonal polynomials and related identities, providing formulas and extending recent mathematical work, with potential applications in analysis and combinatorics.

Contribution

It defines new polynomial families, proves their orthogonality, and derives identities and formulas, expanding the theoretical framework of orthogonal polynomials.

Findings

$ ext{D}_n^{(r)}(x)$ are orthogonal polynomials for } r > -rac{1}{2}

Derived formulas for $d_n^{(r)}(x)^2$ and linearization of products

Extended recent work of Sun and Guo on polynomial identities

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.