Solution to recursion equation: $ T(n) = T(n-1) *T(n-2) $ with $T(1) = a$, $T(2) = b$

Question

Solve the recursive equation

$$ T(n) = T(n-1) T(n-2) $$

with $T(1) = a$, $T(2) = b$

How do I solve this algebraically?

I unrolled the recursion and got a solution of $a^{f_n}b^{f_{n+1}}$, $f_n$ being the $n$th Fibonacci. But I wish to do this algebraically.

Thanks in advance.

Answer 1

If $a$ and $b$ are positive, you can let $U(n)=\log T(n)$, then $U(n)=U(n-1)+U(n-2)$, and it is an ordinary recursion.