D.Bailey, J. Borwein, and D.Bradley found the beautiful pair involving binomial sums. In Theorem 1 of this paper (2008), let x non-zero integer, then,
When x = 0, they reduce into,
However, there is a third single-term equality,
so there might be a third identity that reduces to this as the special case x = 0.
To compare, there are three identities such that as , then those zeta values are the respective limit. For x integer, then,
The first two were found by Leshchiner and Koecher, respectively, while the third is Theorem 2 in the same paper by Bailey, Borwein, and Bradley. The function is given in Mathematica as,
while is the Euler-Mascheroni constant. So are the Bailey-Borwein-Bradley pair of binomial sum identities in fact a triplet?