<< Let C(m,n) be the binomial coefficient which is, I believe, pronounced in English as "m choose n", i.e. C(m,0)=1, C(m,n) = C(m,n-1)*(m-n+1)/n. If a is any complex number, not a nonnegative integer, it is easy to derive the asymptotic behaviour of the absolute value of C(a,n) as n goes to infinity. I did not succeed in finding this formula in the literature, except for a hint in "Analytic function theory" by Markushevich. Could any group member kindly help me with a reference, preferably a rather old one, since this is surely a classical result? (The formula helps in deriving Abel's theorem on convergence of the binomial series on the unit circle.) >>
Not what you are asking, but you might like to look at my article
The binomial coefficient function, American Mathematical Monthly, 103 (January 1996), 1-17. Also see the cover of the August-September issue, 1995.
(The opening figures are very badly reproduced, but I can send a better version to anyone who is interested.)