Two $q$-analogues of Euler's formula $\zeta (2)={\pi }^2\mathrm{/}6$

It is well known that $\zeta (2)={\pi }^2\mathrm{/}6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum _{k=0}^{\mathrm{\infty }}\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1}{)}^2}=\prod _{n=1}^{\mathrm{\infty }}\frac{(1-q^{2n}{)}^4}{(1-q^{2n-1}{)}^4}$$ and $$\sum _{k=0}^{\mathrm{\infty }}\frac{q^{2k-\lfloor (-1{)}^{k}k\mathrm{/}2\rfloor }}{(1-q^{2k+1}{)}^2}=\prod _{n=1}^{\mathrm{\infty }}\frac{(1-q^{2n}{)}^2(1-q^{4n}{)}^2}{(1-q^{2n-1}{)}^2(1-q^{4n-2}{)}^2},$$ where $q$ is any complex number with $\mathrm{\mid }q\mathrm{\mid }<1$. We also give a $q$-analogue of the identity $\zeta (4)={\pi }^4\mathrm{/}90$, and pose a problem on $q$-analogues of Euler’s formula for $\zeta (2m)(m=3,4,\dots )$.