Suppose we have sequences $\{a_i\}_{i=0}^\infty$ and $\{b_i\}_{i=0}^\infty$ that correspond to generating functions $A(x)=\sum_{i=0}^\infty a_ix^i$ and $B(x)=\sum_{i=0}^\infty b_ix^i$.
Let $c\in \mathbb N$ be a fixed constant. If we would have some particular relation among $a_i$ and $b_i$, for example if $b_i=i^ca_i$, or if $b_i=\sum_{j=0}^ia_j $, then we can express $B(x)$ in terms of $A(x)$ easily, using the standard generating function tools like $c$-differentiation and convolution with $\frac{1}{1-x}$, respectively.
However, if we change the mentioned two cases slightly, in particular,
- if $b_i=a_i\cdot\sum_{j=0}^ia_j $, or
- if $b_i=a_i^c$,
for all $i\in\mathbb N_0 $, then expressing $B(x)$ in terms of $A(x)$ seems a much harder task. Any thoughts, hints or suggestions are most welcome. Also, if someone can do it using e.g.f. or some other types of generating functions, feel free to change the definitions of $A$ and $B$.