binomial(x, m) gives incorrect answer when m is float

[ppurka]

Essentially, this stuff

sage: binomial(10, 0.1)
1

What I would have expected it to give as output is

sage: gamma(10+1)/gamma(10-0.1+1)/gamma(0.1+1)
1.32919082253922

This enhancement has to be done on the sage side since both mpir and pari don't give the correct answer. Using the gamma function in these special cases will also take care of complex numbers.

But, we must avoid changing the whole implementation to the gamma function - that was also giving incorrect results and was fixed in #12448. The sage.rings.arith.binomial function itself is a bit of a mess and it is hard to figure out what will eventually get executed in that code.

Component: basic arithmetic

Keywords: binomial

Issue created by migration from https://trac.sagemath.org/ticket/14897