Sorry if this question (inspired by the recent flurry of activity around a "new" formula for $\pi$) is too naive.
Imitating what one does with Hadamard products, one can try to do the same with sums. Consider $\Gamma(x)$. It has poles at $0$, $-1$,... with known residues, so we can write $$\Gamma(x)=\sum_{n\ge0}\dfrac{(-1)^n}{n!(x+n)}+f(x)\;,$$ where $f(x)$ is an entire function. $f$ cannot be $0$ (for instance make $x\to\infty$), and we readily find that $f(x)=\int_1^\infty t^{x-1}e^{-t}\,dt$. Good.
Now let's do the same with the beta function $B(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$ considering $y$ as a fixed parameter. One finds that $$\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}=\sum_{n\ge0}\dfrac{(1-y)_n}{n!(x+n)}+g(x,y)$$ for some function $g(x,y)$ which has no more poles in $x$, where $(a)_n$ is the (rising) Pochhammer symbol. For $x$, $y$ real the series converges only for $y>0$ I believe. Is there a "trivial" way to see that $g(x,y)$ is identically $0$ (if it is indeed $0$) ?
