A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function

Abstract

In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials.

Keywords

• Bernoulli numbers and polynomials
• Euler numbers and polynomials
• Apostol–Bernoulli and Apostol–Euler numbers and polynomials
• Hurwitz–Lerch zeta function
• Hurwitz zeta function
• alternating Hurwitz zeta function