The coefficients of differentiated expansions and derivatives of ultraspherical polynomials

Abstract

A formula expressing the ultraspherical coefficients of the general order derivative of an infinitely differentiable function in terms of its original ultraspherical coefficients is stated in a more compact form and proved in a simpler way than the formula suggested by Karageorghis and Phillips in their recent report [5]. Formulas expressing explicitly the derivatives of ultraspherical polynomials of any degree and for any order in terms of the ultraspherical polynomials are given. The special cases of Chebyshev polynomials of the first and second kinds and of Legendre polynomials are considered. An application of how to use ultraspherical polynomials for solving ordinary and partial differential equations is described. © 1991.