Fibonomial coefficient

Last updated September 13, 2021

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

Special values

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

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

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

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

{\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,

{\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.


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

The recurrence relation

{\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 $\varphi ={\frac {1+{\sqrt {5}}}{2}}$ :

{\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 $G_{n}$ , that is, a sequence that satisfies $G_{n}=G_{n-1}+G_{n-2}$ for every $n,$ then

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

for every integer $n$ , and every nonnegative integer $k$ .

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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.

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