I found this identity online to help simplify the partition function of a harmonic oscillator system:
$$\sum^\infty_{\kappa=0}\begin{pmatrix} \alpha+\kappa-1\\ \kappa \end{pmatrix}z^\kappa=\frac{1}{(1-z)^\alpha},|z|\le1$$
I have searched the internet pretty extensively this morning and have not found any proof or justification for this identity. What is the justification for this identity?