I want to derive formula for generating function $\sum_{n=0}^{+\infty}{m+n\choose m}z^n$ because it is very often very useful for me. Unfortunately I'm stuck:
$ f(z)=\sum_{n\ge 0}{m+n\choose n}z^n= \\ \sum_{n\ge 1}{m+n-1\choose n-1}z^n+\sum_{n\ge 0}{m+n-1\choose n}z^n= \\ zf(z)+\sum_{n\ge 0}{m+n-1\choose n}z^n$
here, I'm afraid that there is a need for something more sophisticated than above trivial identity. Can you help me?