I'd like to prove the following identity:
(The identity referred from : http://dlmf.nist.gov/26.8#E11)
Rather than using induction, I would like to find another way to prove this identity.
I just superficially understands that the generating function has a intrinsic relation to recurrence relation, and what I know about the recurrence upon Stiring the second kind is:
$$S(n,k) = S(n-1,k-1) + k \cdot S(n-1,k)$$
How can I make up the object identity from this recurrence relation?