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

Mehmet Ali Özarslan; Ceren Ustaoğlu; Mehmet Ali Özarslan; Ceren Ustaoğlu

doi:10.3934/era.2022087

Electronic Research Archive

2022, Volume 30, Issue 5: 1723-1747. doi: 10.3934/era.2022087

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

Mehmet Ali Özarslan ^{1
,
,},
Ceren Ustaoğlu ²

1.
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, Northern Cyprus, via Mersin 10, Turkey
2.
Department of Computer Engineering, Final International University, Kyrenia, Northern Cyprus, via Mersin 10, Turkey

Academic Editor: Jean-Daniel Djida

Received: 01 December 2021 Revised: 02 March 2022 Accepted: 06 March 2022 Published: 28 March 2022

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.
Citation: Mehmet Ali Özarslan, Ceren Ustaoğlu. Extended incomplete Riemann-Liouville fractional integral operators and related special functions[J]. Electronic Research Archive, 2022, 30(5): 1723-1747. doi: 10.3934/era.2022087

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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 $ 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.

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