Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2019; 34(2): 523-532

Online first article November 16, 2018      Printed April 30, 2019

https://doi.org/10.4134/CKMS.c180147

Copyright © The Korean Mathematical Society.

Certain integration formulae for the generalized $k$-Bessel functions and Deleure hyper-Bessel function

Yongsup Kim

Wonkwang University

Abstract

Integrals involving a finite product of the generalized Bessel functions have recently been studied by Choi {\it et al.}~\cite{Ch-Ag, Ch-Ku-Pu}. Motivated by these results, we establish certain unified integral formulas involving a finite product of the generalized $k$-Bessel functions. Also, we consider some integral formulas of the $(p, q)$-extended Bessel functions $J_{\nu,p,q}(z)$ and the Delerue hyper-Bessel function which are proved in terms of $(p, q)$-extended generalized hypergeometric functions, and the generalized Wri\-ght hypergeometric functions, respectively.

Keywords: Gamma function, generalized hypergeometric function ${}_pF_q$, generalized (Wright) hypergeometric functions ${}_p\Psi_q$, generalized Lauricella series in several variables, generalized $k$-Bessel function of the first kind, Oberhettinger's integral form

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