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.

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.

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

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

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 ${\displaystyle k}$.

References

• Benjamin, Arthur T.; Plott, Sean S., A combinatorial approach to Fibonomial coefficients (PDF), Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711, archived from the original (PDF) on 2013-02-15, retrieved 2009-04-04CS1 maint: location (link)
• Ewa Krot, An introduction to finite fibonomial calculus, Institute of Computer Science, Bia lystok University, Poland.
• Weisstein, Eric W. "Fibonomial Coefficient". MathWorld.
• Dov Jarden, Recurring Sequences (second edition 1966), pages 30–33.