Find the convolution if OFGs

Question

f(x) = E aj * (x^3)^j ; g(x) = E bj * (x^2)^j

Sorry for the format, with E I meant to type Sigma. How do I find [x^k] (f(x)*g(x)).

Answer 1

Assuming \begin{align*} f(x)=\sum_{j=0}^\infty a_jx^{3j}\qquad \text{and} \qquad g(x)=\sum_{j=0}^\infty b_j x^{2j} \end{align*}

we obtain \begin{align*} [x^k]\left(f(x)\cdot g(x)\right)&=[x^k]\left(\sum_{p=0}^\infty a_px^{3p}\sum_{q=0}^\infty b_q x^{2q}\right)\\ &=[x^k]\left(\sum_{n=0}^\infty \sum_{{3p+2q=n}\atop{p,q\geq 0}} a_pb_q x^n\right)\\ &=\sum_{{3p+2q=k}\atop{p,q\geq 0}} a_pb_q \end{align*}