Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

Abstract

In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s. It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation.

Keywords

• analytic number theory
• λ-generalized Hurwitz–Lerch zeta functions
• derivative properties
• recurrence relations
• integral representations
• Mellin transform