A $q$-analogue for Euler's evaluations of the Riemann zeta function

TL;DR

This paper introduces a $q$-analogue of Euler's formulas for even zeta values, extending recent work and providing a new perspective on classical evaluations of the Riemann zeta function.

Contribution

It presents a novel $q$-analogue for $ ext{zeta}(2k)$, generalizing previous results and expanding the understanding of zeta function evaluations in the $q$-series context.

Findings

Established a $q$-analogue for $ ext{zeta}(2k)$

Generalized Sun's results for $ ext{zeta}(2)$ and $ ext{zeta}(4)$

Provided explicit formulas in Theorems 3.1 and 3.2

Abstract

We provide a $q$-analogue of Euler's formula for $\zeta(2k)$ for $k\in\mathbb{Z}^+$. Our main results are stated in Theorems 3.1 and 3.2 below. The result generalizes a recent result of Z.W. Sun who obtained $q$-analogues of $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$.