Abstract
This paper is concerned with representing sums of the finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations, each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, which involve the hypergeometric functions 1 F 1 and 2 F 1 .
Highlights
We will fix some notations and recall some basic facts about relevant orthogonal polynomials that will be used throughout this paper
The following orthogonalities with respect to various weight functions are enjoyed by Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials
The sums of finite products of Bernoulli, Euler and Genocchi polynomials have been expressed as linear combinations of Bernoulli polynomials in [12,13,14]
Summary
We will fix some notations and recall some basic facts about relevant orthogonal polynomials that will be used throughout this paper. The following orthogonalities with respect to various weight functions are enjoyed by Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials. The sums of finite products of Bernoulli, Euler and Genocchi polynomials have been expressed as linear combinations of Bernoulli polynomials in [12,13,14] These were done by deriving Fourier series expansions for the functions closely related to those sums of finite products. Along the same line as the present paper, sums of finite products of Chebyshev polynomials of the second, third and fourth kinds and of Fibonacci, Legendre and Laguerre polynomials were expressed in terms of all kinds of Chebyshev polynomials in [16,17,18]. We let the reader refer to [19,20] for some applications of Chebyshev polynomials and to [21,22,23,24,25] for some similar iteration methods
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