A $$\varvec{q}$$-Analogue for Euler’s $$\varvec{\zeta (6)=\pi ^6/945}$$

Abstract

Abstract Recently, Sun (Two q-analogues of Euler’s formula $$\zeta (2)=\pi ^2/6$$ζ(2)=π2/6. arXiv:1802.01473, 2018) obtained q-analogues of Euler’s formula for $$\zeta (2)$$ζ(2) and $$\zeta (4)$$ζ(4). Sun’s formulas were based on identities satisfied by triangular numbers and properties of Euler’s q-Gamma function. In this paper, we obtain a q-analogue of $$\zeta (6)=\pi ^6/945$$ζ(6)=π6/945. Our main results are stated in Theorems 2.1 and 2.2 below.