Gamma/Poisson Distribution- Probability

Let be a gamma random variable with parameters . That is, its density is , where is a constant that does not depend on . Suppose also that the conditional distribution of given that is Poisson with mean . That is . Show that the conditional distribution of given that is the gamma distribution with parameters .

So the problem is to find and show that it is the gamma distribution. Now . So I am guessing that the and the are the factors that affect whether the distribution is Poisson or Gamma.

PS: Is R the best stats software in general? Is Jmp good?