A survey on third and fourth kind of Chebyshev polynomials and their applications

Abstract

In this paper, we have a survey on Chebyshev polynomials of third and fourth kind, which are respectively orthogonal with respect to the weight functions ρ 1 ( x ) = ( 1 + x ) 1 / 2 ( 1 - x ) - 1 / 2 and ρ 2 ( x ) = ( 1 - x ) 1 / 2 ( 1 + x ) - 1 / 2 on [−1, 1]. These sequences are special cases of Jacobi polynomials P n ( α , β ) ( x ) for α + β = 0 and appear in the potential theory because of the nature of foresaid case differential equation. General properties of these two sequences such as orthogonality relations, differential equations, recurrence relations, decomposition of sequences, Rodrigues type formula, representation of polynomials in terms of hypergeometric functions, generating functions, their relation with the first and second kind of Chebyshev polynomials, upper and lower bounds and eventually estimation of two definite integrals as ∫ - 1 1 ρ 1 ( x ) f ( x ) d x and ∫ - 1 1 ρ 2 ( x ) f ( x ) d x are represented. Moreover, under the Dirikhlet conditions, an analytic function can be expanded in terms of the Chebyshev polynomials of third and fourth kind. Finally, what distinguishes these two sequences from other orthogonal polynomials is to satisfy a semi minimax property that has application in approximating the functions of type Q ( x ) P n ( x ) where P n ( x ) is an arbitrary polynomial of degree n and Q ( x ) denotes a constant weighting factor. In this way, some numerical examples are also given.