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

TL;DR

This paper extends Ramanujan's identity for odd zeta values to the setting of function fields, involving Bernoulli-Carlitz numbers, and provides a new analogue in this mathematical context.

Contribution

It introduces a novel analogue of Ramanujan's identity for Bernoulli-Carlitz numbers in function fields, expanding the understanding of special value identities in this area.

Findings

Established an identity analogous to Ramanujan's for Bernoulli-Carlitz numbers.

Connected the identity to classical zeta value formulas and Eisenstein series.

Provided a new perspective on special values in function field arithmetic.

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.