A kind of orthogonal polynomials and related identities

Zhi-Hong Sun

paper

A kind of orthogonal polynomials and related identities

arXiv:1606.08327

Abstract

In this paper we introduce the polynomials ${d_{n}^{(r)} (x)}$ and ${D_{n}^{(r)} (x)}$ given by $d_{n}^{(r)} (x) = \sum_{k = 0}^{n} (k x + r + k) (n - k x - r) (n \geq 0)$ , $D_{0}^{(r)} (x) = 1, D_{1}^{(r)} (x) = x$ and $D_{n + 1}^{(r)} (x) = x D_{n}^{(r)} (x) - n (n + 2 r) D_{n - 1}^{(r)} (x) (n \geq 1) .$ We show that ${D_{n}^{(r)} (x)}$ are orthogonal polynomials for $r > - \frac{1}{2}$ , 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.

16 pages

A kind of orthogonal polynomials and related identities

Abstract

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