Fibonomial coefficient

In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as

${\displaystyle {\binom {n}{k}}_{F}={\frac {F_{n}F_{n-1}\cdots F_{n-k+1}}{F_{k}F_{k-1}\cdots F_{1}}}={\frac {n!_{F}}{k!_{F}(n-k)!_{F}}}}$

where n and k are non-negative integers, 0 ≤ k ≤ n, F_j is the j-th Fibonacci number and n!_F is the nth Fibonorial, i.e.

${\displaystyle {n!}_{F}:=\prod _{i=1}^{n}F_{i},}$

where 0!_F, being the empty product, evaluates to 1.

The fibonomial coefficients can be expressed in terms of the Gaussian binomial coefficients and the golden ratio ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$:

${\displaystyle {\binom {n}{k}}_{F}=\varphi ^{k\,(n-k)}{\binom {n}{k}}_{-1/\varphi ^{2}}=(-\varphi )^{k\,(k-n)}{\binom {n}{k}}_{-\varphi ^{2}}.}$

Special values

The Fibonomial coefficients are all integers. Some special values are:

${\displaystyle {\binom {n}{0}}_{F}={\binom {n}{n}}_{F}=1}$

${\displaystyle {\binom {n}{1}}_{F}={\binom {n}{n-1}}_{F}=F_{n}}$

${\displaystyle {\binom {n}{2}}_{F}={\binom {n}{n-2}}_{F}={\frac {F_{n}F_{n-1}}{F_{2}F_{1}}}=F_{n}F_{n-1},}$

${\displaystyle {\binom {n}{3}}_{F}={\binom {n}{n-3}}_{F}={\frac {F_{n}F_{n-1}F_{n-2}}{F_{3}F_{2}F_{1}}}=F_{n}F_{n-1}F_{n-2}/2,}$

${\displaystyle {\binom {n}{k}}_{F}={\binom {n}{n-k}}_{F}.}$

Fibonomial triangle

The Fibonomial coefficients (sequence A010048 in the OEIS) are similar to binomial coefficients and can be displayed in a triangle similar to Pascal's triangle. The first eight rows are shown below.

${\displaystyle n=0}$									1
${\displaystyle n=1}$								1		1
${\displaystyle n=2}$							1		1		1
${\displaystyle n=3}$						1		2		2		1
${\displaystyle n=4}$					1		3		6		3		1
${\displaystyle n=5}$				1		5		15		15		5		1
${\displaystyle n=6}$			1		8		40		60		40		8		1
${\displaystyle n=7}$		1		13		104		260		260		104		13		1

The recurrence relation

${\displaystyle {\binom {n}{k}}_{F}=F_{n-k+1}{\binom {n-1}{k-1}}_{F}+F_{k-1}{\binom {n-1}{k}}_{F}}$

implies that the Fibonomial coefficients are always integers.

Applications

Dov Jarden proved that the Fibonomials appear as coefficients of an equation involving powers of consecutive Fibonacci numbers, namely Jarden proved that given any generalized Fibonacci sequence ${\displaystyle G_{n}}$, that is, a sequence that satisfies ${\displaystyle G_{n}=G_{n-1}+G_{n-2}}$ for every ${\displaystyle n,}$ then

${\displaystyle \sum _{j=0}^{k+1}(-1)^{j(j+1)/2}{\binom {k+1}{j}}_{F}G_{n-j}^{k}=0,}$

for every integer ${\displaystyle n}$, and every nonnegative integer ${\displaystyle k}$.