On fractional integration formulae for Aleph functions

This paper is devoted to the study of a new special function, which is called, according to the symbol used to represent this function, as an Aleph function. This function is an extension of the I-function, which itself is a generalization of the well-known and familiar Gand H-functions in one variable. In this paper, a notation and complete definition of the Aleph function will be presented. Fractional integration of the Aleph functions, in which the argument of the Aleph function contains a factor t(1 t)l, k, l > 0, will be investigated. The results derived are of most general character and include many results given earlier by various authors including Kilbas [10], Kilbas and Saigo [11] and Galué [6] and others. The results obtained form the key formulae for the results on various potentially useful special functions of physical and biological sciences and technology available in the literature. 2011 Elsevier Inc. All rights reserved.