I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version):
$$(1-\alpha )^{-n}=\sum _{x=0}^{\infty } \frac{n}{n+x \beta }\frac{\Gamma(n+x \beta +1)}{\Gamma(x+1)\Gamma(n+x (\beta -1)+1)}\alpha ^x(1-\alpha )^{x(\beta -1)}, \quad 0<\alpha<1.$$
My question is that, apart from Lagrange's formula(see paper), is there any other way to show this identity?
Thanks in advance.