Derivatives of Srivastava's hypergeometric functions with respect to their parameters

Abstract

This paper studies derivatives with respect to the parameters of Srivastava triple hypergeometric functions HA, HB and HC. Using basic properties of the Gamma function and Pochhammer symbols, we obtain explicit formulas for first and higher order derivatives. These derivatives are expressed in terms of Pathan quadruple hypergeometric function F. We also derive Euler type differential operator identities, contiguous relations for unit shifts in the parameters, and recurrence relations satisfied by these derivatives. In addition, we show that derivatives of arbitrary order satisfy systems of linear partial differential equations in the underlying variables. The results extend known differentiation formulas for classical and multivariable hypergeometric functions and provide tools for potential applications in mathematical physics and engineering.