x
F(n + 1,x) - F(n,x) = - ------------------------------ F(n + 2,x),
([gamma] + n)([gamma] + n + 1)
whence we obtain
F(1,x) 1 x/[gamma]([gamma] + 1) x/([gamma] + 1)([gamma] + 2)
------ = -- ---------------------- ----------------------------
F(0,x) 1 + 1 + 1 + ...,
which may also be written
[gamma] x x
------- ----------- -----------
[gamma] + [gamma] + 1 + [gamma] + 2 + ...
By putting +- x squared/4 for x in F(0,x) and F(1,x), and putting at the same
time [gamma] = 1/2, we obtain
x x squared x squared x squared x x squared x squared x squared
tan x = -- -- -- -- tanh x = -- -- -- --
1 - 3 - 5 - 7 - ... 1 + 3 + 5 + 7 + ...
These results were given by Lambert, and used by him to prove that [pi]
and [pi] squared incommensurable, and also any commensurable power of e.