Derivatives of any Horn-type hypergeometric functions with respect to their parameters

Abstract

We consider the derivatives of Horn hypergeometric functions of any number of variables with respect to their parameters. The derivative of such a function of n variables is expressed as a Horn hypergeometric series of n+1 infinite summations depending on the same variables and with the same region of convergence as for the original Horn hypergeometric function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series, and generalized Lauricella series are explicitly presented. Applications to the calculations of Feynman diagrams are discussed, especially the series expansions in ϵ within dimensional regularization. Connections with other classes of special functions are discussed as well.