# Finding Probability generating function from moment generating function

I have been trying to solve a master equation and now finally when I solved it using the method of moment generating function. I don't know how to convert it into a corresponding probability function. The moment generating function $F[x,y,t]$ is of the form of,

$$F[x,y,t] = e^{(k_1 x+k_2 y+\frac{k_3}{\sqrt{xy}}+k_4)t} F[0]$$

Where, $$F[x,y,t]=\sum_{r=0}^{\infty} \sum_{s=0}^{\infty} x^r y^s P[r,s,t]$$ and $P[r,s,t]$ is the probability distribution function that I am looking for and $t$ is time.

What probability generating function does it corresponds to?

Nitin

• What exactly are $x,y,t$? Write how $F$ is defined. – zhoraster Mar 29 '16 at 19:44
• Now this is impossible unless $k_3=0$. In the latter case, the answer is easy and independent of $t$ (as $F$ is). – zhoraster Mar 31 '16 at 19:33