Honsberger's Identity/Negative Indices

Theorem
Let $n \in \Z_{< 0}$ be a negative integer.

Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers).

Then Fibonacci Number in terms of Smaller Fibonacci Numbers:


 * $F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

continues to hold, whether $m$ or $n$ are positive or negative.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\le 0}$, let $\map P n$ be the proposition:
 * $F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

This can equivalently be written:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $F_{m - n} = F_{m - 1} F_{-n} + F_m F_{-n + 1}$

$\map P 0$ is the case:

Thus $\map P 0)$ is seen to hold.

Basis for the Induction
$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ and $\map P {k - 1}$ are true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $F_{m - k} = F_{m - 1} F_{-k} + F_m F_{-k + 1}$

and:
 * $F_{m - \paren {k - 1} } = F_{m - 1} F_{-\paren {k - 1} } + F_m F_{-\paren {k - 1} + 1}$

from which it is to be shown that:

Induction Step
First we note that:

and:

This is the induction step:

So $\map P k \land \map P {k - 1} \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\le 0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$