Fractional derivatives of the H-function of several variables

Abstract

In the present paper we derive a number of key formulas involving fractional derivatives for the H-function of several variables, which was introduced and studied in a series of papers by H. M. Srivastava and R. Panda [cf., e.g., J. Reine Angew. Math. 283/284 (1976), 265–274; J. Reine Angew. Math. 288 (1976), 129–145; Comment. Math. Univ. St. Paul. 24 (1975), fasc. 2, 119–137; ibid. 25 (1976), fasc. 2, 167–197; Nederl. Akad. Wetensch. Proc. Ser. A 81 = Indag. Math. 40 (1978), 118–131 and 132–144; Nederl. Akad. Wetensch. Proc. Ser. A 82 = Indag. Math. 41 (1979), 353–362; see also Bull. Inst. Math. Acad. Sinica 9 (1981) , 261–277]. We make use of the generalized Leibniz rule for fractional derivatives in order to obtain one of the aforementioned results, which involves a product of two multivariable H-functions. Each of these results is shown to apply to yield interesting new results for certain multivariable hypergeometric functions and, in addition, several known results due, for example, to J. L. Lavoie, T. J. Osler and R. Tremblay [ SIAM Rev. 18 (1976) , 240–268], H. L. Manocha and B. L. Sharma [ J. Austral. Math. Soc. 6 (1966), 470–476; J. Indian Math. Soc. (N.S.) 38 (1974) , 371–382] and R. K. Raina and C. L. Koul [ Jñānābha 7 (1977) , 97–105].