I need to check if the following limit is 1 or not. $$\lim_{n, m \to \infty}\frac{\Gamma (n+m+2)}{\Gamma (n+m-x-y+1)}\int_{0}^1\int_{0}^1 p^{n}(1-p)^{n-x}q^{m-1}(1-q)^{m-y-1}dpdq=?$$
where $0<p,q<1$, $0<x<n$, and $0<y<m$.
Could anyone help me? Thanks
By Fubini's Theorem, $$\begin{align*} \int_0^1\int_0^1 p^{n}(1-p)^{n-x}q^{m-1}(1-q)^{m-y-1}dp\,dq &= \left( \int_0^1 p^n (1-p)^{n-x}\,dp \right) \left(\int_0^1 q^{m-1}(1-q)^{m-y-1}\,dq \right) \\ &= \mathrm{B}(n+1,n-x+1) \mathrm{B}(m,m-y) \end{align*}$$ where $\mathrm{B}$ is the B Function. It has a known identity $$ \mathrm{B}(u,v) = \frac{\Gamma(u)\Gamma(v)}{\Gamma(u + v)} $$ Therefore, $$ \mathrm{B}(n+1,n-x+1) \mathrm{B}(m,m-y) = \frac{\Gamma(n+1)\Gamma(n-x+1)\Gamma(m)\Gamma(m-y)}{\Gamma(2n - x + 2)\Gamma(2m - y)} $$