Two $q$-analogues of Euler’s formula $\zeta (2)=\pi ^2/6$
Tom 158 / 2019
Colloquium Mathematicum 158 (2019), 313-320
MSC: Primary 05A30; Secondary 11B65, 11M06, 33D05.
DOI: 10.4064/cm7686-11-2018
Opublikowany online: 5 September 2019
Streszczenie
It is well known that $\zeta (2)=\pi ^2/6$ as discovered by Euler. We present the following two $q$-analogues of this celebrated formula: \begin{align*} \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},\\ \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}, \end{align*} where $q$ is any complex number with $|q| \lt 1$. 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 $\zeta (2m)\ (m=3,4,\ldots )$.
