A Brief Study of Operators of Generalized Fractional Integration

Fractional calculus broadens the scope of conventional calculus by introducing derivatives and integrals of non whole number orders. This mathematical field expands the ideas of differentiation and integration beyond integer values, offering versatile methods for describing intricate processes across diverse scientific and engineering disciplines. The abstract explores the fundamental definitions, properties, and applications of fractional operators, including the Riemann-Liouville, Holmgren, and Grünwald-Letnikov approaches given by different mathematicians like the Mellin transform which have established connections, while a few of them explored the relationships of the Hankel transform. In this survey, ideas from Kiryakov. V was taken especially related to a more unusual instance of kernels that were special functions like the Gauss and generalized hypergeometric functions, including arbitrary G- and H-functions, kernels and to create a theory of the associated GFC with several applications. Additionally, five more authors brought attention to their respective contributions in this area. In this survey, the Riemann-Liouville fractional integral is simplified to the Weyl integral, and a brief study is done on the hypergeometric functions of one and more variables, such as the generalized hypergeometric function contributed by a few mathematicians.