An analogue of Ramanujan's identity for Bernoulli-Carlitz numbers

Abstract

In his second notebook, Ramanujan discovered the following identity for the special values of $\zeta(s)$ at the odd positive integers \begin{equation*}\begin{aligned}\alpha^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n = 1}^{\infty}\dfrac{n^{-2m - 1}}{e^{2\alpha n} - 1}\right\} &-(- \beta)^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n = 1}^{\infty}\dfrac{n^{-2m - 1}}{e^{2\beta n} - 1}\right\}\nonumber &=2^{2m}\sum_{k = 0}^{m + 1}\dfrac{\left(-1\right)^{k-1}B_{2k}\,B_{2m - 2k+2}}{\left(2k\right)!\left(2m -2k+2\right)!}\,\alpha^{m - k + 1}\beta^k \label{(1.2)},\end{aligned} \end{equation*} where $ \alpha $ and $ \beta $ are positive numbers such that $ \alpha\beta = \pi^2 $ and $ m $ is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler's famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan's identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.