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