Prove equation with binomial coefficient

Question

Prove $$\sum_{k=0}^s{\frac{\binom{s}{k}}{\binom{t}{r+k}}} = \frac{t+1}{(t+1-s)\binom{t-s}{r}} $$ for $r + s \le t$

Answer 1

We can apply the beta function identity \begin{align*} \binom{n}{k}^{-1}=(n+1)\int_0^1x^k(1-x)^{n-k}\,dx \end{align*} We start with the left hand side using this identity and then rearrange it and apply the binomial theorem. The right hand side follows.

We obtain \begin{align*} \color{blue}{\sum_{k=0}^s}&\color{blue}{\binom{s}{k}\binom{t}{r+k}^{-1}}\\&=\sum_{k=0}^s\binom{s}{k}(t+1)\int_0^1x^{r+k}(1-x)^{t-r-k}\,dx\\ &=(t+1)\int_{0}^1x^r(1-x)^{t-r}\sum_{k=0}^s\binom{s}{k}\left(\frac{x}{1-x}\right)^k\,dx\\ &=(t+1)\int_{0}^1x^r(1-x)^{t-r}\left(1+\frac{x}{1-x}\right)^s\,dx\\ &=(t+1)\int_0^1x^r(1-x)^{t-s-r}\,dx\\ &\color{blue}{=\frac{t+1}{t-s+1}\binom{t-s}{r}^{-1}} \end{align*} and the claim follows.