On Sums of Special Functions

Abstract

Sums of the form \(\sum_{k=0}^{n}\frac{\prod_{i=1}^{p}\left(a_{i}\right)_{k}}{\prod_{j=1}^{q}\left(b_{j}\right)_{k}}w^{k}{}F_{k}(z)\) are considered\(,\) where \(F_{k}(z)\) are special functions of hypergeometric type. Such sums involving Bessel, Struve, incomplete gamma functions, and Laguerre, Hermite, Jacobi, Legendre, and Chebyshev polynomials can be represented in terms of the Kampé de Fériet and generalized Horn hypergeometric functions of two variables.