Abstract
The principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.
Highlights
Special functions in general and orthogonal polynomials, in particular, have been used for centuries
3 Connection formulas between the sixth-kind Chebyshev polynomials and some orthogonal polynomials
We have established some new formulas concerned with the sixth-kind Chebyshev polynomials
Summary
Special functions in general and orthogonal polynomials, in particular, have been used for centuries. To solve the connection problems between the sixth-kind Chebyshev polynomials and some other orthogonal polynomials, we follow two different approaches. 3.1 Solution of the sixth-kind Chebyshev-ultraspherical connection problem Theorem 1 Let k be any non-negative integer. The sixth-and first-kind Chebyshev polynomials are connected by the following two formulas: Y2k(x) = 22k–1. The sixth- and second-kind Chebyshev polynomials are connected by the following two formulas:. The sixth-kind Chebyshev polynomials and Legendre polynomials are connected by the following two formulas:. 3.2 Solution of the ultraspherical sixth-kind Chebyshev connection problem This section derives the inversion connection formulas to those given in the last subsection. × (3 – 2m)2 + 2k(3 – 4m) Y2k–2m+1(x)
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