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

Abstract

In this paper, we consider sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the previously-studied ones. We represent each of them as linear combinations of Chebyshev polynomials of all kinds whose coefficients involve some terminating hypergeometric functions 2 F 1 . The results may be viewed as a generalization of the linearization problem, which is concerned with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. These representations are obtained by explicit computations.

On the Chebyshev polynomials and some of their new identities

“…Some notes: It is worth noting that Theorem 1 has been obtained by different methods in equation 29of [9], but the expression is different from our result. In fact equation 29in [9] involved the Gauss hypergeometric function, so it looks a little bit more complicated, and our Theorem 1 is simple and straightforward. Theorem 2 has been obtained by different methods in (1.30) of [3].…”

“…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

“…In [27], the following lemma is stated for m ≥ r + 1. But it is valid for any nonnegative integer m, under the usual convention r+1 j = 0, for j > r + 1 (see [21]).…”

“…Then our results for α m,r (x), β m,r (x), and γ m,r (x) will be obtained by making use of Lemmas 1 and 2, the general formulas in Propositions 1 and 2, and integration by parts. We note here that each of the sums in (37)-(39) are also expressed in terms of all four kinds of Chebyshev polynomials in (21).…”

Representing by Orthogonal Polynomials for Sums of Finite Products of Fubini Polynomials

“…Before we close this section, we would like to recall some of the previous results related to the present work. In the same way as in this study, some sums of finite products of Chebyshev polynomials of the first, second, third and fourth kinds, and those of Legendre, Laguerre, Lucas and Fibonacci polynomials are expressed in terms of Chebyshev polynomials of all kinds (see [4,[16][17][18][19]) and also in terms of Hermite, extended Laguerre, Legendre, Gegenbauer and Jacobi polynomials (see [20][21][22][23]).…”