Fibonacci, Lucas, Chebyshev
In this short article I want to point out some striking similarities between Fibonacci numbers, Lucas numbers, and Chebyshev polynomials of second and first kind.
First let me give some basic definitions, which differ slightly from the standard ones, for reasons which will be apparent later:
Definitions
The Fibonacci numbers $F_n$ and the Lucas numbers $F_n$ can be defined as traces.
Consider first the matrix $\Phi$ and its characteristic polynomial $\chi_\Phi(t)$:
$$ \Phi = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \qquad \chi_\Phi(t) = t^2 - t - 1 $$
Then
$$ L_n = \tr\left(\Phi^n\right) \qquad F_n = \tr\left( \frac{\Phi^n}{\chi_\Phi'(\Phi)} \right) $$
for $n \in \ZZ$.
Similarly Chebyshev polynomials of the second kind $\Che_n$ and of the first kind $\Sha_n$ can be defined by
considering the matrix $\Psi$ and its characteristic polynomial $\chi_\Psi(t)$:
$$ \Phi = \begin{pmatrix} 0 & 1 \\ -1 & 2x \end{pmatrix} \qquad \chi_\Phi(t) = t^2 - 2xt + 1 $$
Then
$$ \Sha_n = \tr\left(\Psi^n\right) \qquad \Che_n = \tr\left( \frac{\Psi^n}{\chi_\Psi'(\Psi)} \right) $$
for $n \in \ZZ$.
Note that with the usual definitions the Chebyshev polynomials (let's denote them $T_n$ and $U_n$) are related to those above as follows:
$$ 2T_n = \Sha_n \qquad U_{n-1} = \Che_n $$
An assortment of correspondences
Below I listed some similar identities I found (without proof).
Feel free to find even more connections!
| Chebyshev World | Fibonacci World |
|---|---|
$$ \Che_{n+2} = 2x\Che_{n+1} - \Che_{n} $$ | $$ F_{n+2} = F_{n+1} + F_{n} $$ |
$$ \Sha_{n+2} = 2x\Sha_{n+1} - \Sha_{n} $$ | $$ L_{n+2} = L_{n+1} + L_{n} $$ |
$$ \Che_{n-2} + \Che_{n} + \Che_{n+2} = \Che_3 \Che_n $$ | $$ F_{n-2} + F_{n} + F_{n+2} = L_3 F_{n} $$ |
$$ \Che_{n-1} + \Che_{n+1} = \Che_2 \Che_n $$ | $$ F_{n-1} + F_{n+1} = F_2 L_n $$ |
$$ \Che_{n+1} - \Che_{n-1} = \Sha_n$$ | $$ F_{n+1} + F_{n-1} = L_n $$ |
$$ \Che_{n+1}^2 - \Che_{n-1}^2 = \Che_2 \Che_{2n} $$ | $$ F_{n+1}^2 - F_{n-1}^2 = F_2 F_{2n} $$ |
$$ \gcd(\Che_{a},\Che_{b}) = \Che_{\gcd(a,b)} $$ | $$ \gcd(F_a,F_b) = F_{\gcd(a,b)} $$ |
With $\alpha = \frac{a}{\gcd(a,b)}$, $\beta = \frac{b}{\gcd(a,b)}$, $\gamma = \gcd(a,b)$ positive integers: | |
$$ \lcm(\Che_{a},\Che_{b}) = \begin{cases} \sum_{k=-r}^{r} \Che{\gamma(\beta + 2k)} & \text{ if }\alpha = 2r+1 \\ \sum_{k=0}^{r-1} \Che{\gamma(\beta - 1- 2k)} + \Che{\gamma(\beta+1+2k)} & \text{ if } \alpha = 2r \\ \end{cases} $$ | |
$$ \lcm(F_a, F_b) = \begin{cases} \sum_{k=-r}^{r} F{\gamma(\beta + 2k)} & \text{ if } \alpha = 2r+1 \\ \sum_{k=0}^{r-1} F{\gamma(\beta - 1- 2k)} + F{\gamma(\beta+1+2k)} & \text{ if }\alpha = 2r \\ \end{cases} $$ | |
$$ \binom{n}{k} _{\Che} := \begin{cases} \frac{\Che _{n}\cdots\Che _{n-k+1}}{\Che _{k}\cdots\Che _{1}} & \text{ if } 0 \leq k \leq n \\\\ 0 & \text{ otherwise} \\ \end{cases} $$ | $$ \binom{n}{k} _{F} := \begin{cases} \frac{ F _{n}\cdots F _{n-k+1}}{ F _{k}\cdots F _{1}} & \text{ if } 0 \leq k \leq n \\\\ 0 & \text{ otherwise} \\ \end{cases} $$ |
$$ \chebinom{n}{k} = \Che_{k+1} \chebinom{n-1}{k} - \Che_{n-k-1} \chebinom{n-1}{k-1} $$ | $$ \fibonom{n}{k} = F_{k+1}\fibonom{n-1}{k} + F_{n-k-1}\fibonom{n-1}{k-1} $$ |
$$ \Psi^n = \begin{pmatrix} -\Che_{n-1} & \Che_{n} \\ -\Che_{n} & \Che_{n+1} \\ \end{pmatrix} $$ | $$ \Phi^n = \begin{pmatrix} F_{n-1} & F_{n} \\ F_{n} & F_{n+1} \\ \end{pmatrix} $$ |
$$ \Che_n^2 - \Che_{n-1}\Che_{n+1} = 1 $$ | $$ F_{n-1}F_{n+1} - F_{n}^2 = (-1)^n $$ |
$$ \vdots $$ | $$ \vdots $$ |
References
- Common definition of Chebyshev polynomials
- R. Knott's collection of Fibonacci formulae
- Rayes, Trevisan, Wang - 'Factorization properties of chebyshev polynomials' (Theorem 4 proves the gcd property)
- My initial tweet on this topic
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