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

TL;DR

This paper derives formulas for the derivatives of Horn hypergeometric functions with respect to parameters, expressing them as higher-dimensional hypergeometric series, with applications to Feynman diagram calculations.

Contribution

It provides explicit formulas for derivatives of Horn hypergeometric functions, expanding their applicability and linking them to Feynman diagram computations.

Findings

Derivatives expressed as Horn hypergeometric series of n+1 variables

Explicit formulas for Appell, Lauricella, and generalized hypergeometric functions

Applications to epsilon-expansion in dimensional regularization

Abstract

We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ 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 original Horn 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 calculation of Feynman diagrams are discussed, especially the series expansion in $\epsilon$ within dimensional regularization. Connections with other classes of special functions are discussed as well.