mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as :

\binom_F = \frac = \frac

where ''n'' and ''k'' are non-negative integers, 0 ≤ ''k'' ≤ ''n'', ''F_j'' is the ''j''-th Fibonacci number and ''n''!_F is the ''n''th

Fibonorial In mathematics, the Fibonorial , also called the Fibonacci factorial, where is a nonnegative integer, is defined as the product of the first positive Fibonacci numbers, i.e. : _F := \prod_^n F_i,\quad n \ge 0, where is the th Fibonacci number, ...

, i.e. :

_F := \prod_^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: :

\binom_F = \binom_F = 1

\binom_F = \binom_F = F_n

\binom_F = \binom_F = \frac = F_n F_,

\binom_F = \binom_F = \frac = F_n F_ F_ /2,

\binom_F = \binom_F.

Fibonomial triangle

The Fibonomial coefficients are similar to

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s and can be displayed in a triangle similar to Pascal's triangle. The first eight rows are shown below. The recurrence relation :

\binom_F = F_ \binom_F  + F_ \binom_F

implies that the Fibonomial coefficients are always integers. The fibonomial coefficients can be expressed in terms of the

Gaussian binomial coefficients In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or ''q''-binomial coefficients) are ''q''-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as \binom n ...

and the golden ratio

\varphi=\frac2

: :

_F = \varphi^_

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_ + G_

for every

n,

then :

\sum_^(-1)^\binom_F G_^k = 0,

for every integer

n

, and every nonnegative integer

k

References

* * Ewa Krot
''An introduction to finite fibonomial calculus''
Institute of Computer Science, Bia lystok University, Poland. * {{MathWorld, title=Fibonomial Coefficient, urlname=FibonomialCoefficient * Dov Jarden, ''Recurring Sequences'' (second edition 1966), pages 30–33. Fibonacci numbers Factorial and binomial topics Triangles of numbers