Zeta functions interpolating the convolution of the Bernoulli polynomials

TL;DR

This paper establishes nonlinear relations among multiple Hurwitz-Riemann zeta functions and extends Euler's nonlinear relation to generalized Bernoulli polynomials and numbers through analytic continuation.

Contribution

It introduces new nonlinear relations for multiple Hurwitz-Riemann zeta functions and generalizes Euler's relation to Bernoulli polynomials and numbers.

Findings

Proves nonlinear relations on multiple Hurwitz-Riemann zeta functions

Extends Euler's relation to generalized Bernoulli polynomials and numbers

Uses analytic continuation to establish these results

Abstract

We prove nonlinear relation on multiple Hurwitz-Riemann zeta functions. Using analytic continuation of these multiple Hurwitz-Riemann zeta function, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers.