Integral representations of the derivatives of the Gauss hypergeometric function with respect to its parameters

Abstract

We derive a generalized expression of the derivatives of the Gauss hypergeometric function with respect to its parameter by introducing an auxiliary function and performing some operations. The obtained expression involves infinite series that is only analytic inside the unit circle |z| = 1. Using the concept of analytic continuation, we obtain an integral representation of the derivatives of the Gauss hypergeometric function that is analytic in the whole complex plane, except at its singularities.