Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Abstract

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

On some analytic properties of tempered fractional calculus

“…Other, more recently developed, models of fractional calculus appear to bear no relationship to Riemann-Liouville whatsoever, beyond the superficial similarity of being defined by an integral transform with a kernel function [8,34,2,31,18,19,27,9,28,29]. However, in these cases too, relationships with the Riemann-Liouville model have been discovered, often in the form of infinite convergent series [4,13].…”

On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

“…Fractional calculus and its applications have been intensively investigated for a long time by many researchers in numerous disciplines and attention to this subject has grown tremendously. By making use of the concept of the fractional derivatives and integrals, various extensions of them have been introduced [27][28][29][30], and authors have gained different perspectives in many areas such as engineering, physics, economics, biology, statistics [31,32]. One of the generalizations of fractional derivatives is Riemann-Liouville k-fractional derivative operator studied in [24,25,33].…”

Extended incomplete version of hypergeometric functions

“…It can be seen easily when y → 1 in equation (8), this equation will reduce to the well-known beta function. In terms of the incomplete beta function B y (x, z), the incomplete Pochhammer ratios b, c; y n and b, c; y n were introduced as follows [10]:…”