How to find a close form expression in terms generating functions for the triple summation

Question

Given $$\sum\limits_{i=0}^\infty a_i z^i=A(z)$$ and $$\sum\limits_{i=1}^\infty b_i z^i=B(z)$$ and $$\sum\limits_{i=0}^\infty c_i z^i=C(z)$$ Find $\sum\limits_{i=1}^\infty\sum\limits_{j=0}^\infty a_j \sum\limits_{k=1}^\infty b_kc_{k+j-i} z^i$ in terms of the generating functions $A(z),B(z),C(z)$.

Answer 1

Since $i=(j)+(k)-(j+k-i)$, the triple sum is $$\sum\limits_{\alpha,\beta,\gamma}a_\alpha b_\beta c_\gamma z^{\alpha+\beta-\gamma}=A(z)B(z)C(1/z).$$