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Two $q$-analogues of Euler's formula $ζ(2)=π^2/6$ (1802.01473v6)

Published 5 Feb 2018 in math.NT and math.CO

Abstract: It is well known that $\zeta(2)=\pi2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum_{k=0}\infty\frac{qk(1+q{2k+1})}{(1-q{2k+1})2}=\prod_{n=1}\infty\frac{(1-q{2n})4}{(1-q{2n-1})4}$$ 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 $\zeta(4)=\pi4/90$, and pose a problem on $q$-analogues of Euler's formula for $\zeta(2m)\ (m=3,4,\ldots)$.

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