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 Fibonorial , also called the Fibonacci factorial, where is a

nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

, 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, and gives the empty product (defined as the

multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

, i.e. 1). The Fibonorial is defined analogously to the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

. The Fibonorial numbers are used in the definition of

Fibonomial coefficient In 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'', ''Fj'' is the ''j''-th Fibonacci ...

s (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients.

Asymptotic behaviour

The series of fibonorials is

asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

to a function of the golden ratio

\varphi

n!_F \sim C \frac

. Here the ''fibonorial constant'' (also called the ''fibonacci factorial constant'')

C

is defined by

C = \prod_^\infty (1-a^k)

, where

a=-\frac

and

\varphi

is the golden ratio. An approximate truncated value of

C

is 1.226742010720 (see for more digits).

Almost-Fibonorial numbers

Almost-Fibonorial numbers: . Almost-Fibonorial primes:

prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

among the almost-Fibonorial numbers.

Quasi-Fibonorial numbers

Quasi-Fibonorial numbers: . Quasi-Fibonorial primes:

among the quasi-Fibonorial numbers.

Connection with the q-Factorial

The fibonorial can be expressed in terms of the

q-factorial In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...

and the golden ratio

\varphi=\frac2

: :

n!_F = \varphi^ \,!.

Sequences

Product of first nonzero Fibonacci numbers . and for such that and are primes, respectively.

References

* {{MathWorld, Fibonorial Fibonacci numbers fr:Analogues de la factorielle#Factorielle de Fibonacci