Complete recurrence relation

Question

There are many questions about $k$-order recurrence relations and answers on how to solve them at the forum. But is there a thing as a "complete" or "full" recurrence relation (I don't know their name)? I mean a recurrence relation where the $n$-th term depends on all that came before, like $$ y_{n+1} = a_n y_n + \cdots + a_0y_0. $$ How could such recurrence be solved?

Answer 1

Assuming $(a_n)_{n \ge 0}$ is a given sequence, and assuming it is the same sequence at each step: $y_{n+1}=a_n y_n+(a_{n-1}y_{n-1}+ \cdots + a_0y_0)=a_ny_n + y_n = (a_n+1)y_n\,$. Then, by telescoping, $y_n = y_0 \cdot \prod_{k=0}^{n-1} (a_k+1)\,$.