A new family of Lerch-type zeta functions interpolating a certain class of higher-order Apostol-type numbers and Apostol-type polynomials

Abstract

The aim of this paper is to investigate some classes of higher-order Apostol-type numbers and Apostol-type polynomials. We construct Lerch-type zeta functions which interpolate these numbers and polynomials at negative integers. Moreover, by combining some well-known identities such as the Chu-Vandermonde identity with the Lerch-type zeta functions and generating functions for the higher-order Apostol-type numbers and Apostol-type polynomials, we derive some relations and identities including functional equation for these Lerch-type zeta functions with other zeta type functions, Raabe-type multiplication formula for the higher-order Apostol-type polynomials and the Stirling numbers. Finally, we give some remarks and observations on Lerch-type zeta functions and their functional equations.

Mathematics Subject Classification (2010): Primary 11B68, 11S40, 11S80, 33E20; Secondary 26C05, 30B40, 44A20.

Keywords: Bernoulli numbers and Bernoulli polynomials, Euler numbers and Euler polynomials, Riemann and Hurwitz (or generalized) zeta functions, Hurwitz-Lerch zeta function, Lerch zeta function, polylogarithm function, multiplication formula, functional equation, Mellin transformation