Extended incomplete Riemann-Liouville fractional integral operators and related special functions

In this study, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operators and investigate their analytical properties rigorously. More precisely, we investigate their transformation properties in $ L_{1} $ and $ L_{\infty} $ spaces, and we observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, definitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. Then the extended incomplete $ \tau $-Gauss, confluent and Appell's hypergeometric functions are introduced by means of the extended incomplete beta functions and some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. As a particular advantage of the new fractional integral operators, some generating relations of linear and bilinear type for extended incomplete $ \tau $-hypergeometric functions have been derived.