Reference for tail bound of chi square distribution

Question

Fisher's method allows to combine a set of p values $p_1,\ldots,p_n$. In order to compute the p value associated with Fisher's method one needs to compute the tail of a chi square distribution with even degrees of freedom. I have seen for instance here: Combining p-values for averaging technical protein quantification replicates in python, see the comment by @whuber, that it is possible to compute this tail as follows: $$ x\sum_{k=0}^{n-1}\frac{(-\log x)^k}{k!} $$ where $x=\prod_{k=1}^{n}p_k$. In fact deriving this identity is not very hard, it just entails a few integration by parts. However, I imagine that this is a well known trick, can someone point me to the appropriate article or textbook?