This is a part of my research. I just need asymptotic value when $k\to \infty$ of $$\frac{\Gamma(k+1)}{\Gamma(k+\zeta+2)}\Bigg\{\frac{\Gamma(n+k+1)\Gamma(k+\zeta+2)}{\Gamma(n+k+\zeta+2)\Gamma(k+1)}-\frac{\Gamma(k-n+\zeta+2)\Gamma(k+1)}{\Gamma(k-n+1)(k+\zeta+2)}\Bigg\}.$$ I tried to use a formula $$\frac{\Gamma (k+a)}{\Gamma (k+b)}\sim k^{a-b}\Big\{1+\frac{1}{2k}(a-b)(a+b-1)+\frac{1}{12k^2}\binom{a-b}{2}\Big(3(a+b-1)^2-(a-b+1)\Big)+O(k^{-3})\Big\}$$
whenever $k\to \infty$. Please do some steps for me then I will follow other easily.