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

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 .

Representation by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first, third and fourth kinds

“…W (r) m+r (x). In fact, these equalities can easily be seen by differentiating the generating functions in (9), (11) and (12). The next three theorems are the main results in this paper.…”

“…Lastly, we would like to mention some of the previous results that are related to the present work. Along the same line as this paper, certain sums of finite products of Chebyshev polynomials of the first, second, third and fourth kinds, and of Legendre, Laguerre, Fibonacci and Lucas polynomials are expressed in terms of all four kinds of Chebyshev polynomials in [10,16,19,23,25] and also in terms of Hermite, extended Laguerre, Legendre, Gegenbauer and Jacobi polynomials in [4,11,13,24].…”

The Power Sums Involving Fibonacci Polynomials and Their Applications

“…In addition, regarding many orthogonal polynomials and famous sequences, Kim et al have done a lot of important research work, obtaining a series of interesting identities. Interested readers can refer to References [16][17][18][19][20][21][22]; we will not list them one by one.…”

Representing by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials

“…Along the same line as this paper, certain sums of finite products of Chebyshev polynomials of the first, second, third, and fourth kinds, and of Legendre, Laguerre, Fibonacci, and Lucas polynomials are expressed in terms of all four kinds of Chebyshev polynomials in [11,16,19,24,26]. Also, certain sums of finite products of Chebyshev polynomials of the second, third, and fourth kinds, and of Fibonacci, Legendre, and Laguerre polynomials are expressed in terms of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials in [5,12,23,27]. Also, we would like to remark here that some Appell and non-Appell polynomials are also expressed as linear combinations of Bernoulli polynomials.…”