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An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions

In this paper we investigate the extension of the multiple Erd elyi-Kober fractional integral operator of Kiryakova to arbitrary complex values of parameters by the way of regularization. The regularization involves derivatives of the function in question and the integration with respect to a kernel expressed in terms of special case of Meijer's G function. An action of the regularized multiple Erd elyi-Kober operator on some simple kernels leads to decomposition formulas for the generalized hypergeometric functions. In the ultimate section, we de ne an alternative regularization better suited for representing the Bessel type generalized hypergeometric function p􀀀1Fp. A particular case of this regularization is then used to identify some new facts about the positivity and reality of zeros of this function.