Abstract
The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions {}_{2}F_{1}, {}_{2}F_{2}, and {}_{1}F_{1}. These representations are obtained by explicit computations.
Highlights
Introduction and preliminariesHere before stating the necessary basic facts about orthogonal polynomials, we will first fix some notations that will be used throughout this paper
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
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 in terms of Hermite, extended Laguerre, Legendre, Gegenbauer and Jacobi polynomials in [4, 11, 13, 24]
Summary
Introduction and preliminariesHere before stating the necessary basic facts about orthogonal polynomials, we will first fix some notations that will be used throughout this paper. The above-mentioned polynomials are given, in terms of generating functions, in the following: 1 – xt F1(t, x) = 1 – 2xt + t2 We will consider the following sums of finite products of Chebyshev polynomials of the first, third and fourth kinds: αm,r(x) = Our results for αm,r(x), βm,r(x), and γm,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.
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