Generating Function in Physics

Generating functions for what?

Sometimes it's an elegant technique to define special functions by a generating function. E.g., the Legendre Polynomials can be defined by
[tex]\frac{1}{\sqrt{r-2 r u+1}}=\sum_{l=0}^{\infty} \mathrm{P}_l(u) r^l.[/tex]
This implies that
[tex]\mathrm{P}_l(u)=\left (\frac{1}{l!} \frac{\mathrm{d}^l}{\mathrm{d} r^l} \frac{1}{\sqrt{r-2 r u+1}} \right)_{r=0}.[/tex]
Do you mean such examples?

 

I believe (and I could be mistaken) that the thermodynamic canonical and grand canonical partition functions are generating functions in the probabilistic sense.