A New Family of the λ -Generalized Hurwitz-Lerch Zeta Functions with Applications

Abstract: Motivated largely by a number of recent investigations, we introduce and investigate the various properties of a certain new family of the λ -generalized Hurwitz-Lerch zeta functions. We derive many potentially useful results involving these λ -generalized Hurwitz-Lerch zeta functions including (for example) their partial differential equations, new series and Mellin-Barnes type contour integral representations (which are associated with Fox’s H-function) and several other summation formulas. We discuss their potential application in Number Theory by appropriately constructing a seemingly novel continuous analogue of Lippert’s Hurwitz measure. We also consider some other statistical applications of the family of the λ -generalized Hurwitz-Lerch zeta functions in probability distribution theory.