J. Borwein and D. Bradley found this can be generalized to . Define the functions,
and so on. Beautiful, aren’t they? Notice that all the (excepting ) are all divisible by 4. This infinite family has a generating function. Let z non-zero integer, then,
On the other hand, for s = 4n+1,
with this version for found by Jim Cullen. There are various versions for both s = 4n+1 and 4n+3. For example, for , we have the relations,
Eliminating the last two terms will yield the shorter relation given by Borwein and Bradley. There is a generating function for all s = 2n+1, but none is known that is only for s = 4n+1. See Apery-Like Formulae for for more details.