  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.
