CM Evaluations of the Goswami-Sun Series

TL;DR

This paper explores special values of Goswami-Sun $q$-series at CM points, revealing their algebraic nature and explicit evaluations involving gamma function ratios, extending Ramanujan's classical formulas.

Contribution

It establishes that the series' values at CM points are algebraic multiples of gamma function ratios, providing explicit evaluations and connecting to Ramanujan's formulas.

Findings

Values at CM points are algebraic multiples of gamma ratios.

Explicit evaluations involve gamma(1/4)^4/π^3 at specific q-values.

Connects series evaluations to classical gamma function identities.

Abstract

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 $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. Goswami extended these series to an infinite family of $q$-series, which he analogously used to obtain new derivations of the evaluations of $\zeta(2k)\in\mathbb{Q}\cdot\pi^{2k}$ for every positive integer $k$. Since it is well known that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$, it is natural to seek further specializations of these series which involve special values of the $\Gamma$-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points $\tau$, where $q:=e^{2\pi i\tau}$, are algebraic multiples of specific ratios of $\Gamma$-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of $\Gamma\left(\frac{1}{4}\right)^4/\pi^3$ when $q=e^{-\pi}$, $e^{-2\pi}$.