On the Generalization of Hypergeometric and Confluent Hypergeometric Functions and Their Applications for Finding the Derivatives of the Generalized Jacobi Polynomials

摘要

Recently, some generalizations of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been introduced in the literature. Most of the special functions, such as Jacobi polynomials, can be expressed in terms of Gauss hypergeometric function (GHF) and confluent hypergeometric function (CHF). The main object of this paper is to express explicitly the derivatives of generalized Jacobi polynomials in terms of Jacobi polynomials themselves, by using generalized hypergeometric functions of any degree that have been differentiated an arbitrary numbers of times. The results for the special cases of generalized Ultraspherical polynomials and Chebyshev polynomials of the first, second, third and fourth kinds and Legendre polynomials are also given.