Fibonomial coefficient

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

[math]\displaystyle{ \binom{n}{k}_F = \frac{F_nF_{n-1}\cdots F_{n-k+1}}{F_kF_{k-1}\cdots F_1} = \frac{n!_F}{k!_F (n-k)!_F} }[/math]

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.

[math]\displaystyle{ {n!}_F := \prod_{i=1}^n F_i, }[/math]

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

Special values

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

[math]\displaystyle{ \binom{n}{0}_F = \binom{n}{n}_F = 1 }[/math]

[math]\displaystyle{ \binom{n}{1}_F = \binom{n}{n-1}_F = F_n }[/math]

[math]\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}, }[/math]

[math]\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, }[/math]

[math]\displaystyle{ \binom{n}{k}_F = \binom{n}{n-k}_F. }[/math]

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.

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

The recurrence relation

[math]\displaystyle{ \binom{n}{k}_F = F_{n-k+1} \binom{n-1}{k-1}_F + F_{k-1} \binom{n-1}{k}_F }[/math]

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 [math]\displaystyle{ \varphi=\frac{1+\sqrt5}2 }[/math]:

[math]\displaystyle{ {\binom n k}_F = \varphi^{k\,(n-k)}{\binom n k}_{-1/\varphi^2} }[/math]

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 [math]\displaystyle{ G_n }[/math], that is, a sequence that satisfies [math]\displaystyle{ G_n = G_{n-1} + G_{n-2} }[/math] for every [math]\displaystyle{ n, }[/math] then

[math]\displaystyle{ \sum_{j = 0}^{k+1}(-1)^{j(j+1)/2}\binom{k+1}{j}_F G_{n-j}^k = 0, }[/math]

for every integer [math]\displaystyle{ n }[/math], and every nonnegative integer [math]\displaystyle{ k }[/math].

References

• Benjamin, Arthur T.; Plott, Sean S., A combinatorial approach to Fibonomial coefficients, Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711, http://www.math.hmc.edu/~benjamin/papers/Fibonomial.pdf, retrieved 2009-04-04 
• Ewa Krot, An introduction to finite fibonomial calculus, Institute of Computer Science, Bia lystok University, Poland.
• Weisstein, Eric W.. "Fibonomial Coefficient". http://mathworld.wolfram.com/FibonomialCoefficient.html. 
• Dov Jarden, Recurring Sequences (second edition 1966), pages 30–33.