Binet Form
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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
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