how to prove a combinatorial identity

By using Lagrange's expansion \begin{align} \frac{f(z)}{1 - w \phi(z)} = \sum_{n=0}^{\infty} \frac{w^{n}}{n!} \, \left. D_{z}^{n} \left\{ f(z) \, [ \phi(z) ]^{n} \right\} \right|_{z = z_{0}} \end{align} where $z = z_{0} + w \phi(z)$ then one easily developes \begin{align} \sum_{n=0}^{\infty} \binom{a+bn}{n} \, \left(\frac{z}{(1+z)^b}\right)^{n} = \frac{(1+z)^{1+a}}{1+(1-b)z} \end{align}