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

Daeyeoul Kim,Yilmaz Simsek

doi:10.3390/math9030233

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.

Highlights

The families of zeta functions and special numbers and polynomials have been studied widely in many areas
After investigating some properties of these functions, we found that these functions can interpolate the Apostol–Bernoulli polynomials, the Bernoulli numbers and polynomials, and the Euler numbers and polynomials at negative integers
By applying the Mellin transformation to the generating function (10), we arrive at the following integral representation of a new family of zeta type functions Z (s; k, a): Z (s; k, a) =

Summary

Introduction

The families of zeta functions and special numbers and polynomials have been studied widely in many areas They have been used to model real-world problems. With the help of the Equation (7), a few values of the Euler polynomials of the first kind are given as follows: 2x − 1. A few of the values of the Euler numbers of the first kind are given as follows: E0 = 1, E1 = − , E2 = 0, E3 = , E5 = − , E7 =. The Euler numbers of the second kind E∗j are given by means of the following generating function: tj. The alternating Hurwitz zeta function can be used to interpolate these numbers at negative integers. Euler eta function (cf. [4,8,31,34])

A New Family of Zeta Type Functions

Conclusions