Abstract
Here, we consider the sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then, we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials, which involve the hypergeometric function 3F2.
Highlights
Introduction and PreliminariesWe first recall here that, for any nonnegative integer n, the falling factorial polynomials ( x )n and the rising factorial polynomials < x >n are respectively given by:( x ) n = x ( x − 1) · · · ( x − n + 1), ( n ≥ 1), ( x )0 = 1, (1)< x > n = x ( x + 1) · · · ( x + n − 1), < x >0 = 1. (2)The two factorial polynomials are related by:( x )n = (−1)n < − x >n,< x >n = (−1)n (− x )n
The Chebyshev polynomials of the first, second, third and fourth kinds are respectively defined by the following generating functions
The general formulas in Proposition 1 can be derived by using orthogonalities and Rodrigues’ formulas for Chebyshev polynomials and integration by parts
Summary
We first recall here that, for any nonnegative integer n, the falling factorial polynomials ( x )n and the rising factorial polynomials < x >n are respectively given by:. The Chebyshev polynomials of the first, second, third and fourth kinds are respectively defined by the following generating functions. We will consider the sums of finite products of Chebyshev polynomials of the third and fourth kinds in (25) and (26). The general formulas in Proposition 1 can be derived by using orthogonalities and Rodrigues’ formulas for Chebyshev polynomials and integration by parts. In [7], the sums of finite products of Chebyshev polynomials in (25) and (26) were expressed as linear combinations of Bernoulli polynomials. The same has been done for the sums of finite products of Bernoulli, Euler and Genocchi polynomials in [8,9,10] All of these were found by deriving. For some other applications of Chebyshev polynomials, we let the reader refer to [11,12,13]
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