Connection Problem for Sums of Finite Products of Legendre and Laguerre Polynomials

Kyung-Won Hwang,Dmitry Dolgy,Dae Kim,Taekyun Kim

doi:10.3390/sym11030317

Kyung-Won Hwang, Dmitry Dolgy + Show 2 more

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https://doi.org/10.3390/sym11030317

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Abstract

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 .

Highlights

In this paper, we will consider two sums of finite products γn,r ( x ) =Pi1 ( x ) Pi2 ( x ) . . . Pi2r+1 ( x ),(n, r ≥ 0), i1 +···+i2r+1 =n (32)in terms of Legendre polynomials and ε n,r ( x ) = i1 +···+ir+1 =n L i1 x r+1 L i2 x r+1
In [16,17,18], sums of finite products of Bernoulli, Euler and Genocchi polynomials were represented as linear combinations of Bernoulli polynomials
Let γm,r ( x ), ε m,r ( x ), and αm ( x ) denote the following sums of finite products given by γn,r ( x ) =

Summary

Preliminaries

After fixing some notations that will be needed throughout this paper, we will review briefly some basic facts about orthogonal polynomials relevant to our discussion. All the facts stated here can be found in [3,4,5,6,7,8].Interested readers may refer to [1,2,9,10,11,12,13] for full accounts of orthogonal polynomials and to [14,15] for papers discussing relevant orthogonal polynomials. The above-mentioned orthogonal polynomials are given, in terms of generating functions, by. For Legendre, Gegenbauer and Jacobi polynomials, we have Rodrigues’ formulas, and for Hermite and generalized Laguerre polynomials, we have Rodrigues-type formulas. The orthogonal polynomials in Equations (22)–(26) satisfy the following orthogonality relations with respect to various weight functions

Introduction

Proof of Theorem 1

Proof of Theorem 2

Conclusions