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.

Highlights

  • Introduction and PreliminariesWe first fix some notations that will be used throughout this paper

  • We see that the two factorial sequences are related by: (−1)n ( x )n =< − x >n

  • We considered the expression (26) in [5] and (27) and (28) in [6] and were able to express each of them in terms of the Chebyshev polynomials of all four kinds

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Summary

Introduction and Preliminaries

We first fix some notations that will be used throughout this paper. For any nonnegative integer n, the falling factorial sequence ( x )n and the rising factorial sequence < x >n are respectively given by:. In terms of generating functions, the Chebyshev polynomials of the first, second, third, and fourth kinds are respectively given by: Symmetry 2018, 10, 742; doi:10.3390/sym10120742 www.mdpi.com/journal/symmetry. We consider the expressions αm,r ( x ), β m,r ( x ), and γm,r ( x ) in (23)–(25), which are sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, respectively. (see Lemmas 2 and 3) by making use of the generating function in (6) This is unlike the previous works for (26)–(28) (see [5,6]), where we showed they are respectively equal to. 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. The reader may want to look at [19,20,21] for some applications of Chebyshev polynomials

Proof of Theorem 1
Proofs of Theorems 2 and 3
Conclusions
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