Communications of the
Korean Mathematical Society
CKMS

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

Article

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Commun. Korean Math. Soc. 2017; 32(2): 287-304

Online first article April 17, 2017      Printed April 30, 2017

https://doi.org/10.4134/CKMS.c150227

Copyright © The Korean Mathematical Society.

Incomplete extended Hurwitz-Lerch zeta functions and associated properties

Rakesh K. Parmar and Ram K. Saxena

Government College of Engineering and Technology, Jai Narain Vyas University

Abstract

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [\emph{Integral Transforms Spec. Funct.} 23 (2012), 659--683] by means of the incomplete Pochhammer symbols $ \left(\lambda; \kappa\right)_{\nu} $ and $ \left[\lambda; \kappa \right]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.

Keywords: gamma functions, incomplete gamma functions, Pochhammer symbol, incomplete Pochhammer symbols, incomplete generalized hypergeometric functions, incomplete Fox-Wright function, generalized Hurwitz-Lerch zeta function, incomplete Hurwitz-Lerch zeta function

MSC numbers: Primary 11M06, 11M35, 33B15, 33C60; Secondary 11B68, 33C65, 33C90