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

Key words:

• 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