Prove that
Combining toghether the well known expressions...
(1)
... and...
(2)
... we obtain...
(3)
Setting t=1 in (3) we arrive to see that is...
(4)
The integral in (4) can probably be 'attacked' using the residue theorem... if we demonstrate that it does'nt contain the term the goal is realized...
Kind regards
Some years ago in an other 'matemathical challenge' I had to find the 'generating function' of the sequence...
(1)
After some effort I arrived to the identity...
(2)
I don't think that (2) in very important for some application, but in any case it is interesting. As most of series of this type in converges for , and that means that (2) is defined also for , where the argument of cosh (*) is imaginary...
Kind regards
Unfortunately, is not very nice to compute.
It turns out that , where is the Modified Bessel Function of the First Kind
If you look at the identities for , it becomes evident that showing the irrationality of is a fairly difficult thing to do.
(Also, check out the last identity listed in the link. )