New Results of the Fifth-Kind Orthogonal Chebyshev Polynomials

Abstract

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions.

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

“…The fifth-kind Chebyshev polynomials are so named because, like the more common fourkind Chebyshev polynomials, they may be written in trigonometric form. In reality, the following trigonometric representation [38] holds for all integers r:…”

“…This trigonometric representation is extremely useful because it enables us to define polynomials C i (x) for negative integers. More precisely, the following identity holds [38]:…”

“…In his PhD thesis, Masjed-Jamei extracted these polynomials from certain generalized symmetric polynomials that involve four parameters. Abd-Elhameed and Youssri, in [38] derived some new formulas concerned with these polynomials. Furthermore, the fifth kind of Chebyshev polynomials were recently used in [39] to treat some FDEs.…”

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

“…There are six classes of Chebyshev polynomials: first, second, third, fourth, fifth, and sixth kinds (Masjed-Jamei 2006). There are many old and recent studies interested in these polynomials (Abd-Elhameed et al 2016;Türk and Codina 2019;Abd-Elhameed andYoussri 2018, 2019;Abd-Elhameed 2021;Abd-Elhameed and Alkhamisi 2021;Atta et al 2022a). In this paper, our main focus is on the first type of Chebyshev polynomials and their shifted ones.…”

Factoring Variants of Chebyshev Polynomials With Minimal Polynomials Of

“…In 2018, shifted Chebyshev polynomials of the fifth kind were used to solve problems involving fractional-order differential equations [3]. In 2021, connection formulas and other properties of the polynomials of the fifth [2] and sixth [4] kinds were given. Also, Sadri and Aminikhah defined two-variable shifted Chebyshev polynomials of the sixth kind of the form in [14, Equation (4.7)], and used them to solve fractional-order partial differential equations numerically.…”