Apery gave,

J. Borwein and D. Bradley found this can be generalized to . Define the functions,

then,

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.

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