CM Evaluations of the Goswami-Sun Series

In recent work, Sun constructed two $q$-series, and he showed that their limits as $q\rightarrow1$ give new derivations of the Riemann-zeta values $ζ(2)=π^2/6$ and $ζ(4)=π^4/90$. Goswami extended these series to an infinite family of $q$-series, which he analogously used to obtain new derivations of the evaluations of $ζ(2k)\in\mathbb{Q}\cdotπ^{2k}$ for every positive integer $k$. Since it is well known that $Γ\left(\frac{1}{2}\right)=\sqrtπ$, it is natural to seek further specializations of these series which involve special values of the $Γ$-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points $τ$, where $q:=e^{2πiτ}$, are algebraic multiples of specific ratios of $Γ$-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of $Γ\left(\frac{1}{4}\right)^4/π^3$ when $q=e^{-π}$, $e^{-2π}$.

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