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?
The Gibbs Free Energy is regarded as a Generating Function for the other thermodynamic properties if it can be expressed as a function of T and P. If this functionality is known, the Gibbs Free Energy can be used to calculate the specific volume, the enthalpy, the internal energy, and the entropy. Unfortunately, there is no convenient experimental method for directly measuring G as a function of T and P.
I believe (and I could be mistaken) that the thermodynamic canonical and grand canonical partition functions are generating functions in the probabilistic sense.
