6
RSolveis the function you're looking for.
The second example evaluates to
Z = RSolveValue[{z[i + 1] == b z[i] + (1 - b) y[i], z[1] == 0}, z , i]
(*Function[{i}, b^(-1 + i) (\!\(\*UnderoverscriptBox[\(\[Sum]\), \(K[1] = 0\), \(\(-1\) +i\)]\(-\((\((\(-1\) + b)\)\ \*SuperscriptBox[\(b\), \(-K[1]\)]\ y[K[1]])\)\)\) -y[0] + b y[0])]*)
Z[1]//Simplify
(*0*)
...
Only top voted, non community-wiki answers of a minimum length are eligible
