Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the set of partitions $\mathcal{B}$ in which the parts divisible by 4 occur at most once, can I multiply the 2 generating functions together to obtain the generating function for the intersection of $\mathcal{A}$ and $\mathcal{B}$ (i.e for the partitions that satisfy both requirements)? And more importantly, is this a technique that I can use for arbitrary such conditions?
The reason I ask is because my homework assignment contains several questions which require me to give the generating function for partitions that have some property AND some other property, and I wanted to know if this technique was a good/valid one to use for these problems.
Also, is there a similar technique for giving the generating function for integer partitions that have some property OR some other property?
Thanks!