New Results of the Fifth-Kind Orthogonal Chebyshev Polynomials

Waleed Mohamed Abd-Elhameed,Seraj Omar Alkhamisi

doi:10.3390/sym13122407

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.

Highlights

Fifth-Kind Orthogonal ChebyshevThe special functions in general and orthogonal polynomials, in particular, are related to a large number of problems in different disciplines such as approximation theory, theoretical physics, chemistry, and some other mathematical branches
The moments formulas are useful in the numerical treatment of ordinary differential equations with polynomials coefficients; The linearization coefficients are useful in the numerical treatment of some non-linear differential equations as followed in [15]; The connection coefficients are very useful in investigating the convergence analysis as followed in [11]
A class of Chebyshev orthogonal polynomials was investigated from a theoretical point of view

Summary

Introduction

The special functions in general and orthogonal polynomials, in particular, are related to a large number of problems in different disciplines such as approximation theory, theoretical physics, chemistry, and some other mathematical branches. It is expected that these formulas will be useful in some applications This gives us another motivation to investigate theoretically the fifth-kind Chebyshev polynomials. It is worth pointing out here that another motivation for our interest in developing the presented formulas in this paper is that these formulas may be useful in some applications Some of their expected uses are listed as follows: The moments formulas are useful in the numerical treatment of ordinary differential equations with polynomials coefficients; The linearization coefficients are useful in the numerical treatment of some non-linear differential equations as followed in [15]; The connection coefficients are very useful in investigating the convergence analysis as followed in [11].

Some Properties and Essential Formulas

Moments Formulas of Chebyshev Polynomials of the Fifth-Kind

Linearization Formulas of Chebyshev Polynomials of the Fifth-Kind

Connection Formulas with Some Orthogonal Polynomials

Conclusions