Английская Википедия:Fibonomial coefficient

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

<math>\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>{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>\binom{n}{0}_F = \binom{n}{n}_F = 1</math>

<math>\binom{n}{1}_F = \binom{n}{n-1}_F = F_n</math>

<math>\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>\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>\binom{n}{k}_F = \binom{n}{n-k}_F.</math>

Fibonomial triangle

The Fibonomial coefficients Шаблон: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>n=0</math>									1
<math>n=1</math>								1		1
<math>n=2</math>							1		1		1
<math>n=3</math>						1		2		2		1
<math>n=4</math>					1		3		6		3		1
<math>n=5</math>				1		5		15		15		5		1
<math>n=6</math>			1		8		40		60		40		8		1
<math>n=7</math>		1		13		104		260		260		104		13		1

The recurrence relation

<math>\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>\varphi=\frac{1+\sqrt5}2</math>:

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

<math>\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>n</math>, and every nonnegative integer <math>k</math>.

References

• Шаблон:Citation
• Ewa Krot, An introduction to finite fibonomial calculus, Institute of Computer Science, Bia lystok University, Poland.
• Шаблон:MathWorld
• Dov Jarden, Recurring Sequences (second edition 1966), pages 30–33.