Investigations in fractional calculus and certain special functions

Abstract

Fractional Calculus is the area that attends to the integration and differentiation of arbitrary order that could be rational, irrational or complex. The thesis investigates and establishes the concept of fractional calculus with certain special functions. newlineThe first chapter is the introductory chapter. In this chapter, history of Fractional calculus, various special functions like Gamma function, Beta function, Mittag-leffler function, Multiindex Mittag-leffl H-function, I-function, F-function of Hartley and Lorenzo, R-function, Dzrbashjan-Gelfond-Leontiev (D-G-L) operators, Riemann-Liouville Fractional Calculus, Weyl Fractional Calculus, Fractional qCalculus etc. are elaborated. newlineThe second chapter is the Literature review that elaborates the extensive work that had been carried out in fractional calculus, fractional q-calculus and prescribed functions by considerable large number of authors over the years. newlineThe third chapter comprises the investigation involved in obtaining several integrals involving the Riemann-Liouville fractional calculus and F-function. newlineIn the fourth chapter, relations between the Weyl fractional integral and Multiindex Dzrbashjan-Gelfond-Leontiev (D-G-L) differentiation and integration with Multiindex Mittag-leffler function are established. newlineFifth chapter illustrates theorems which connect the Weyl fractional integral to the product of Multiindex Mittag-Lefler function with I-function and with H-function. newlineChapter six addresses the relationship connecting the fractional q-derivation with Multiindex Dzrbashjan-Gelfond-Leontiev (D-G-L) operators for differentiation and integration and the R-function. newlineAs special cases, we obtained some very interesting and well known results studied by prominent authors. newline newline