Differentiation formulas of some hypergeometric functions with respect to all parameters

In this work we present two methods to derive some differentiation formulas of the generalized hypergeometric function mFn(a1,…,am;b1,…,bn;z)mFn(a1,…,am;b1,…,bn;z), including the most commonly used Gauss hypergeometric function 2F1(μ,ν;λ;z)2F1(μ,ν;λ;z) and Kummer confluent hypergeometric function 1F1(μ;ν;z)1F1(μ;ν;z) as special cases, with respect to all parameters. We first briefly describe the direct derivative method for the convergent power series of hypergeometric functions. Secondly, we mainly focus on the differential equation method, which is based on differentiating the generalized hypergeometric differential equation with respect to parameters. Particularly, by using the differential equation method, some general analytical expressions of any sth derivatives with respect to single parameter can be deduced by induction in s. Moreover, we can obtain all the mixed derivatives of higher order very conveniently. Finally, examples are given to illustrate the usefulness of these derivatives in mathematics, physics and other related fields. Numerical examples for computing those singular oscillatory integrals presented in Kang et al. (2013) and Kang and Ling (in press), in turn verify that the approximation value of the required derivatives can be of great precision, and show the correctness of differentiation formulas obtained by the proposed methods.