The computation of $\zeta(2k)$, $\beta(2k+1)$ and beyond by using telescoping series

TL;DR

This paper introduces simple proofs for special values of the Riemann zeta and Dirichlet beta functions using telescoping series and Bernoulli and Euler polynomials, also deriving related integral formulas.

Contribution

It provides novel, straightforward proofs for classical series formulas and extends the method to series involving Apostol polynomials and complex shifts.

Findings

Simplified proofs for (2k) and (2k+1) using telescoping series.

Derivation of integral formulas for (2k+1) and (2k).

Extension of the method to series over integers with complex shifts.

Abstract

We present some simple proofs of the well-known expressions for \[ \zeta(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad \beta(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $\zeta(2k+1)$ and $\beta(2k)$. In addition, the method also applies to series of type \[ \sum_{m\in\mathbb{Z}} \frac{1}{(2m-\mu)^s}, \qquad \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-\mu)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials.