Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2018; 55(2): 599-610

Online first article January 12, 2018      Printed March 30, 2018

https://doi.org/10.4134/BKMS.b170193

Copyright © The Korean Mathematical Society.

Fractional Integration and differentiation of the $(p, q)$-extended Bessel function

Junesang Choi, Rakesh K. Parmar

Dongguk University, Government College of Engineering and Technology

Abstract

We aim to present some formulas for Saigo hypergeometric fractional integral and differential operators involving $(p, q)$-extended Bessel function $J_{\nu,\,p,q} (z)$, which are expressed in terms of Hadamard product of the $(p, q)$-extended Gauss hypergeometric function and the Fox-Wright function $_{p}\Psi_{q}(z)$. A number of interesting special cases of our main results are also considered. Further, it is emphasized that the results presented here, which are seemingly complicated series, can reveal their involved properties via those of the two known functions in their respective Hadamard product.

Keywords: $(p, q)$-extended Bessel function, $(p,q)$-extended hypergeometric function, Fox-Wright function, fractional calculus operators, Hadamard product

MSC numbers: Primary 26A33, 33B20, 33C20; Secondary 26A09, 33B15, 33C05