The work of H. M. Srivastava (9) on generalizations of an interesting identity of Carlson is expanded upon in order to obtain series identities analogous to all four of the Appell functions and their generalizations. Reduction formulas for some Kampé de Fériet functions are derived as consequences of the various double-series identities considered in this paper.
Recently T. R. Prabhakar used fractional integrals in order to obtain explicit solutions to a convolution integral equation in which the kernel involved a confluent hypergeometric function. Decomposition of the integral operator into fractional integrals and exponential functions plays a role in the development and, following the ideas of A. Erd61yi, this decomposition is treated here in a clearer format from the standpoint of Mikusifiski operators. Further, the conditions for existence and uniqueness of the solution are conveniently displayed.In a recent paper by T. R. Prabhakar [4] fractional integrals are used in order to obtain explicit solutions of a convolution integral equation of the formin which the kernel is of the form K(a; b; 20 [tb-/F(b)]lF(a; b; 2t) and 1Fa denotes the confluent hypergeometric function with Re b > 0. Decomposition of the operator Ka, into a product of simpler operators plays a role in the development. Following the ideas of A. Erd61yi [2], this equation can be studied from the standpoint of Mikusifiski calculus and the decompositions can be presented in a clearer format.The following correspondences to Mikusifiski operators are needed" K(a;b;2t)s"-b(s-2) -, Reb>0, Re2>0;It"-et/F(#)] (s 2)-", Re # > 0.In view of the development by Erd61yi [2], these can be obtained from tables of Laplace transforms [3]. Since by Kummer's transformation, 1Fa(a;b; -20=e-Xt F(b-a'b'20 we see that the first relation also holds for Re 2 < 0. For 2 0 in the second correspondence we obtain l" s -u, where I" denotes the Riemann-Liouville fractional integral operator of order / for Re/ > 0. This can be extended to Re/ =< 0 in the usual manner by choosing a positive integer n such that Re(n +/0 > 0 and letting I" I-"I "+" in which I-" denotes a differentiation operator such that the property I-"f s"fis retained.We note that -(d) f (u) du -(s 2)-"f.
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