Prove properties of the factorial (gamma function)

Question

I want to prove the equation is satisfied. 'p' is a natural number. $$\sum_{n=1}^{∞} \frac{\Gamma(n)}{\Gamma(n+p+1)}=\frac{1}{p^2\Gamma(p)}$$ Understandably, This formula can be written as this. $$\sum_{n=1}^{∞} \frac{1}{n(n+1)\cdots(n+p)}=\frac{1}{pp!}$$ Please give me hint or prove.

Answer 1

Here is an approach. Recalling the function $\frac{1}{1-x}$, then the series in consideration can related to it by noticing it is the fraction integral of order $p+1$

$$ \sum_{n=1}^{∞} \frac{\Gamma(n)}{\Gamma(n+p+1)}x^{n+p}=\sum_{n=0}^{∞} \frac{\Gamma(n+1)}{\Gamma(n+p+2)}x^{n+p+1}\\=\frac{1}{\Gamma(p+1)}\int_{0}^{x}(x-t)^{p}\frac{1}{1-t}dt. $$

Now, just evaluate the last integral and then substitute $x=1$. See a related problem.

Note: It is easier to substitute $x=1$ in the integral and then evaluate it, that's is

$$ \frac{1}{\Gamma(p+1)}\int_{0}^{1}(1-t)^{p}\frac{1}{1-t}dt =\dots\,.$$

Answer 2

Have you looked at the function $$ F_{p}(x)=\sum_{n=1}^{\infty}\frac{x^{n+p}}{n(n+1)\cdots(n+p)},\;\;\; |x| < 1 ? $$ The value you want is $\lim_{x\uparrow 1}F_{p}(x)$.

Answer 3

The Beta function to the rescue:

$$\frac{\Gamma(n)\Gamma(p+1)}{\Gamma(n+p+1)} = B(n,p+1) = \int_0^1 t^{n-1}(1-t)^p\,dt.$$