@article{Özarslan2022, 
author = {Mehmet Ali Özarslan and Ceren Ustaoğlu},
title = {Extended incomplete Riemann-Liouville fractional integral operators and related special functions},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {5},
pages = {1723-1747},
keywords = {generating functions, extended incomplete hypergeometric functions, extended incomplete Appell's functions, incomplete fractional calculus},
url = {https://www.sciopen.com/article/10.3934/era.2022087},
doi = {10.3934/era.2022087},
abstract = {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  L1 and  L∞ 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  τ-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  τ-hypergeometric functions have been derived.}
}