Incomplete Pochhammer Ratios and Related Special Functions

This thesis includes five chapters. In the first chapter, general information and some preliminaries that used throughout the thesis are given. In Chapter 2, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function ( ) With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss hypergeometric, confluent hypergeometric and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas and recurrence relation. Furthermore, an incomplete Riemann-Liouville fractional derivative operators are introduced. This definition plays a key role for our understanding of the linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions. In Chapter 3, we give the definitions of Caputo fractional derivative operators and show their use in the special function theory. For this purpose, new types of incomplete hypergeometric functions are introduced and their integral representations are obtained. Furthermore, we define incomplete Caputo fractional derivative operators and show that the images of some elementary functions under the action of incomplete Caputo fractional derivative operators give a new type of incomplete hypergeometric functions. For the new type incomplete hypergeometric functions linear and bilinear generating relations are obtained. In Chapter 4, generalizations of incomplete gamma, beta, Gauss, confluent and Appell’s hypergeometric functions are introduced. Also, Mellin transforms, transformation formulas, differentiation and difference formulas and fractional calculus formulas are obtained for these functions. In Chapter 5, the extended incomplete Mittag-Leffler functions are introduced by using the extended incomplete beta functions and we investigate several properties of these functions. The Mellin transform of these functions is presented with regards to the incomplete Wright hypergeometric functions. Furthermore, we obtain the images of the extended incomplete Mittag-Leffler functions under the actions of Riemann-Liouville fractional integral and derivative operator. Some miscellaneous properties of these funtions are also given. Keywords: incomplete Pochhammer ratios, incomplete hypergeometric functions, incomplete Riemann-Liouville fractional derivative operators, Generating functions, incomplete Caputo fractional derivative operators, generalized incomplete gamma and beta functions, generalized incomplete hypergeometric functions, incomplete Mittag-Leffler functions, extended incomplete Mittag-Leffler functions