Binet Form

Theorem

Let $m \in \R$.

Define:

\(\ds \Delta\)

\(=\)

\(\ds \sqrt {m^2 + 4}\)

\(\ds \alpha\)

\(=\)

\(\ds \frac {m + \Delta} 2\)

\(\ds \beta\)

\(=\)

\(\ds \frac {m - \Delta} 2\)

First Form

The recursive sequence:

$U_n = m U_{n - 1} + U_{n - 2}$

where:

\(\ds U_0\)

\(=\)

\(\ds 0\)

\(\ds U_1\)

\(=\)

\(\ds 1\)

has the closed-form solution:

$U_n = \dfrac {\alpha^n - \beta^n} \Delta$

Second Form

The recursive sequence:

$V_n = m V_{n - 1} + V_{n - 2}$

where:

\(\ds V_0\)

\(=\)

\(\ds 2\)

\(\ds V_1\)

\(=\)

\(\ds m\)

has the closed-form solution:

$V_n = \alpha^n + \beta^n$

where $\Delta, \alpha, \beta$ are as for the first form.

Also see

• Relation Between First and Second Form of Binet Form

Source of Name

This entry was named for Jacques Philippe Marie Binet.

Sources

• Weisstein, Eric W. "Binet Forms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinetForms.html