>>371
Vacca Euler's constant で検索すると、下記ヒット
下記の式(3)あたりか
ありがとう
しかし、Euler's constantは、まだ（2023年でも）無理数であることの証明もまだなんだね

https://cs.uwaterloo.ca/journals/JIS/VOL13/Pilehrood/pilehrood2.pdf
Journal of Integer Sequences, Vol. 13 (2010),
Vacca-Type Series for Values of the
Generalized Euler Constant Function
and its Derivative
Khodabakhsh Hessami Pilehrood1 and Tatiana Hessami Pilehrood2
Institute for Studies in Theoretical Physics and Mathematics
Tehran, Iran

Abstract
We generalize well-known Catalan-type integrals for Euler’s constant to values of
the generalized Euler constant function and its derivatives. Using generating functions
appearing in these integral representations, we give new Vacca and Ramanujan-type
series for values of the generalized Euler constant function and Addison-type series for
values of the generalized Euler constant function and its derivative.

（こっちはちょっと古いが類似か）
https://arxiv.org/abs/math/0508042
New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)
Jonathan Sondow
[v1] Mon, 1 Aug 2005 16:25:05 UTC (44 KB)