linguistic:paraphrasing, quality:corruption, quality:correction
Identities to be used in the proof

The following hold for any complex number α:

Unless α {\displaystyle \alpha } is a nonnegative integer (in which case the binomial coefficients vanish as k {\displaystyle k} is larger than α {\displaystyle \alpha }), a useful asymptotic relationship for the binomial coefficients is, in Landau notation:

This is essentially equivalent to Euler's definition of the Gamma function:

and implies immediately the coarser bounds

for some positive constants m and M.

Formula (2) for the generalized binomial coefficient can be rewritten as