Derivatives of Srivastava’s triple hypergeometric functions

We derive explicit closed-form expressions for the first-order derivatives with respect to all numerator and denominator parameters of Srivastava’s triple hypergeometric functionsHA,HB andHC. By differentiating the defining triple-series termwise and using standard identities for the Gamma and digamma functions together with Pochhammer-symbol relations, each parameter derivative is represented as a finite linear combination of Pathan’s quadruple hypergeometric function F(4) P with appropriately shifted parameter arrays. This unified F(4) P - based description provides a convenient framework for contiguous relations under unit shifts of the parameters and for sensitivity calculations. We also record Euler-type operator identities in the variables (x, y, z) and deduce the resulting systems of linear partial differential equations satisfied by HA, HB and HC. Numerical experiments based on truncated triple-series evaluation are included to corroborate the formulas and to illustrate parameter dependence; we specify the truncation bounds, finite-difference step sizes, and residual-type control criteria used in the computations. MSC Classification: 33C20 , 33C65 , 11J72