Sum over k of n Choose k by Fibonacci t to the k by Fibonacci t-1 to the n-k by Fibonacci m+k
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Theorem
- $\ds \sum_{k \mathop \ge 0} \binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$
where:
- $\dbinom n k$ denotes a binomial coefficient
- $F_n$ denotes the $n$th Fibonacci number.
Proof
The proof proceeds by induction on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $\forall m, t \in \N: \ds \sum_{k \mathop \ge 0} \binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$
$\map P 0$ is the case:
- $\ds \binom 0 0 {F_t}^0 {F_{t - 1} }^0 F_{m + 0} = F_m$
Thus $\map P 0$ is seen to hold.
Basis for the Induction
$\map P 1$ is the case:
\(\ds \binom 1 0 {F_t}^0 {F_{t - 1} }^1 F_{m + 0} + \binom 1 1 {F_t}^1 {F_{t - 1} }^0 F_{m + 1}\) | \(=\) | \(\ds F_{t - 1} F_m + F_t F_{m + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds F_{m + t}\) | Honsberger's Identity |
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 $\map P q$ is true, where $q \ge 1$, then it logically follows that $\map P {q + 1}$ is true.
So this is the induction hypothesis:
- $\forall m, t \in \N: \ds \sum_{k \mathop \ge 0} \binom q k {F_t}^k {F_{t - 1} }^{q - k} F_{m + k} = F_{m + t q}$
from which it is to be shown that:
- $\forall m, t \in \N: \ds \sum_{k \mathop \ge 0} \binom {q + 1} k {F_t}^k {F_{t - 1} }^{q + 1 - k} F_{m + k} = F_{m + t \paren {q + 1} }$
Induction Step
This is the induction step:
\(\ds F_{m + t \paren {q + 1} }\) | \(=\) | \(\ds F_{m + t q} F_{t - 1} + F_{m + t q + 1} F_t\) | Honsberger's Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom q k {F_t}^k {F_{t - 1} }^{q - k} F_{m + k} F_{t - 1} + \sum_{k \mathop \ge 0} \binom q k {F_t}^k {F_{t - 1} }^{q - k} F_{m + 1 + k} F_t\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom q k {F_t}^k {F_{t - 1} }^{q + 1 - k} F_{m + k} + \sum_{k \mathop \ge 0} \binom q k {F_t}^{k + 1} {F_{t - 1} }^{q - k} F_{m + 1 + k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom q k {F_t}^k {F_{t - 1} }^{q + 1 - k} F_{m + k} + \sum_{k \mathop \ge 1} \binom q {k - 1} {F_t}^k {F_{t - 1} }^{q - \paren {k - 1} } F_{m + k}\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom q k {F_t}^k {F_{t - 1} }^{q + 1 - k} F_{m + k} + \sum_{k \mathop \ge 0} \binom q {k - 1} {F_t}^k {F_{t - 1} }^{q - k + 1} F_{m + k}\) | $\dbinom q {-1} = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \paren {\binom q k + \binom q {k - 1} } {F_t}^k {F_{t - 1} }^{q + 1 - k} F_{m + k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \binom {q + 1} k {F_t}^k {F_{t - 1} }^{q + 1 - k} F_{m + k}\) | Pascal's Rule |
So $\map P q \implies \map P {q + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall m, n, t \in \N: \ds \sum_{k \mathop \ge 0} \binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$
$\blacksquare$
Also see
The following are corollaries to this theorem:
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $23$