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

Abstract

In this paper, we investigate sums of finite products of Chebyshev polynomials of the first kind and those of Lucas polynomials. We express each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials whose coefficients involve some terminating hypergeometric functions 1 F 1 and 2 F 1. These are obtained by means of explicit computations.

On the Chebyshev polynomials and some of their new identities

“…We all know that the polynomials T n (x) and U n (x) play important roles in the study of orthogonality of functions and approximation theory, so many scholars have studied their properties and obtained a series of valuable research results. In particular, in the references we have seen that Kim and his team have done a lot of important research work (see [3][4][5][6][7][8][9][10][11]), and Cesarano (see [12][13][14]) has also made a lot of contributions. Some other papers related to these polynomials and sequences can be found in references [2,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

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].…”

“…We studied Eq. (40) in [13,16] and (41) and (42) in [4,19] and were able to express each of them in terms of the Chebyshev polynomials of all kinds, Hermite polynomials, extended Laguerre polynomials. Legendre polynomials, Gegenbauer polynomials, and Jacobi polynomials.…”

Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

“…In this regard, the authors in [16] represented certain sums of finite products of Chebyshev polynomials in terms of Chebyshev polynomials. The authors in [17] represented sums of finite products of Chebyshev polynomials of the first-kind and Lucas polynomials in terms of some orthogonal polynomials. Other expressions of sums of finite products of Legendre and Laguerre polynomials were developed in [18].…”