Let $\{c_{m,n}\}_{m,n\in\mathbb{N}}$ be known complex numbers. My question is, how to find all number series $\{a_{n}\}_{n\in\mathbb{N}}$ such that $$a_n=\sum_{m=0}^\infty c_{m,n}a_{m+n},~\forall n\in\mathbb{N}?$$
For some special cases, the question is easy. For example, if $$c_{m,n}=\begin{cases}0,&m=0,\\ 1.&m>0,\end{cases}$$ then $$a_n=\sum_{m=1}^\infty a_{m+n},~\forall n\in\mathbb{N},$$ so $$a_n=\sum_{m=1}^\infty a_{m+n}=a_{n+1}+\sum_{m=1}^\infty a_{m+n+1}=a_{n+1}+a_{n+1}=2a_{n+1},~\forall n\in\mathbb{N}$$ and we can easily conclude that $$a_{n}=\frac{a_0}{2^n},~\forall n\in\mathbb{N}.$$ In general the simplification like above does not work, e.g., maybe the following problem is hard to solve: $$a_n=\sum_{m=0}^\infty \binom{m+n}{m}a_{m+n},~\forall n\in\mathbb{N}.$$
So is there any way to tackle all such problems? Or like Diophantine equations, only special cases can be tackled?