Mathematically, you were almost there:
F(n) &= f(F(n+1)) & n \in \mathbb Z, \ a \leq n < b, \\
F(b) &= 0.
That's all a mathematician would need to see.
The value of $F(a)$ is completely defined by this,
although this requires the reader to actually think about how to produce the value of $F(a)$ given the value of $F(b).$
If you want to encode explicit step-by-step instructions,
you can write the algorithm in the form of a computer program.
If you want that to be a mathematical expression, you could try to learn denotational semantics, but that probably wouldn't actually make anything better.