Needless to say: If f(z,w) is defined as f(z,w) = gamma(z+1) / (gamma(w+1) * gamma(z-w+1)) then it's a ratio of meromorphic functions, therefore meromorphic. But of course since beta(z,w) = gamma(z) gamma(w) / (gamma(z+w) then we have f(z,w) = 1 / beta(w+1, z-w+1) (and by symmetry = 1 / beta(z-w+1, w+1) ) —Dan ----- Why stop at two? There are a denumerable number of singularities of Gamma(n+1) / Gamma(k+1) Gamma(n-k+1) at negative integer n , and once you decide to start tinkering with approaching them from multiple randomly selected directions, a whole continuum of choices opens before you --- though admittedly not all of them yield integer results. [ And bear in mind, incidentally, that you will have royally screwed up your symbolic theorem prover --- see examples from Mathematica and Maple in my note. ] WFL On 7/23/17, James Propp <jamespropp@gmail.com> wrote:
Maybe n-choose-k should actually be two-valued (like sqrt(z)), with a branch-point at n=k=0?