Some new identities of Chebyshev polynomials and their applications

Abstract

In this paper, we use the properties of Chebyshev polynomials, elementary methods, and combinational techniques to study the computational problem of one kind of convolution sums involving second kind Chebyshev polynomials, and we give an exact computational method, which expresses the sums as second kind Chebyshev polynomials. As some applications of our results, we also obtain several new identities and congruences involving the second kind Chebyshev polynomials, Fibonacci numbers, and Lucas numbers. MSC: 11B39

Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions

“…I the main interest of the paper [4] is to represent the derivatives of U n .x/ in terms of the Chebyshev polynomials themselves. To this aim an "exact computational method" (a recursion formula) was presented.…”

Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials

“…By differentiating equation (1.5) it was shown in [15] and mentioned in [13] that the sum of products in (1.9) can be neatly expressed as in the following. This will play a crucial role in this paper.…”

“…It is well known that the Chebyshev polynomials of the second kind are explicitly given by (see [11, 13]) The r th derivative of (2.1) is given by Then, combining (2.1) and (2.3), we obtain …”

“…The Chebyshev polynomials of the first kind, the Chebyshev polynomials of the second kind, and the Fibonacci polynomials are respectively defined by the recurrence relations as follows (see [13–15]): When , () is the Fibonacci sequence.…”

“…From (1.1), (1.2), and (1.3), it can be easily shown that the generating functions for , , and are respectively given by (see [13–15]): …”