Sums of finite products of Legendre and Laguerre polynomials

Abstract

In this paper, we study sums of finite products of Legendre and Laguerre polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we are going to express those sums of finite products as linear combinations of Bernoulli polynomials. Further, by using a method other than Fourier series expansions, we will be able to express those sums in terms of Euler polynomials.

Representation by Chebyshev Polynomials for Sums of Finite Products of Chebyshev Polynomials

“…In terms of Bernoulli polynomials, quite a few sums of finite products of some special polynomials are expressed. They include Chebyshev polynomials of all four kinds, and Bernoulli, Euler, Genocchi, Legendre, Laguerre, Fibonacci, and Lucas polynomials (see [10][11][12][13][14][15][16]). All of these expressions in terms of Bernoulli polynomials have been derived from the Fourier series expansions of the functions closely related to each such polynomials.…”

“…All of these expressions in terms of Bernoulli polynomials have been derived from the Fourier series expansions of the functions closely related to each such polynomials. Further, as for Chebyshev polynomials of all four kinds and Legendre, Laguerre, Fibonacci, and Lucas polynomials, certain sums of finite products of such polynomials are also expressed in terms of all four kinds of Chebyshev polynomials in [5,6,17,18]. Finally, the reader may want to look at [19][20][21] for some applications of Chebyshev polynomials.…”

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

“…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].…”

Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

“…Chebyshev polynomials are diversely used in approximation theory and numerical analysis, Hermite polynomials appear as the eigenfunctions of the harmonic oscillator in quantum mechanics, Laguerre polynomials have important applications to the solution of Schrödinger's equation for the hydrogen atom, Legendre polynomials can be used to write the Coulomb potential as a series, Gegenbauer polynomials play an important role in the constructive theory of spherical functions and Jacobi polynomials occur in the solution to the equations of motion of the symmetric top. All the necessary facts on those special polynomials can be found in [1][2][3][4][5][6][7][8][9]. For the full accounts of this fascinating area of orthogonal polynomials, the reader may refer to [10][11][12][13].…”

“…n (x), and P (α,β) n (x). Also, the connection problem of expressing in terms of all kinds of Chebyshev polynomials were done for sums of finite products of Chebyshev polynomials of the second, third and fourth kinds and of Fibonacci, Legendre and Laguerre polynomials in [19][20][21].…”