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We consider the derivatives of Horn hypergeometric functions of any numbervariables with respect to their parameters. The derivative of the function in$n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinitesummations depending on the same variables and with the same region ofconvergence as for original Horn function. The derivatives of Appell functions,generalized hypergeometric functions, confluent and non-confluent Lauricellaseries and generalized Lauricella series are explicitly presented. Applicationsto the calculation of Feynman diagrams are discussed, especially the seriesexpansion in $\epsilon$ within dimensional regularization. Connections withother classes of special functions are discussed as well.

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