# Fibonorial

In mathematics, the Fibonorial n!F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive Fibonacci numbers, i.e.

${\displaystyle {n!}_{F}:=\prod _{i=1}^{n}F_{i},\quad n\geq 0,}$

where Fi is the ith Fibonacci number, and 0!F gives the empty product (defined as the multiplicative identity, i.e. 1).

The Fibonorial n!F is defined analogously to the factorial n!. The Fibonorial numbers are used in the definition of Fibonomial coefficients (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 to a function of the golden ratio ${\displaystyle \phi }$: ${\displaystyle n!_{F}\sim C{\frac {\phi ^{n.(n+1)/2}}{5^{n/2}}}}$,

where the fibonorial constant ${\displaystyle C}$ is defined by ${\displaystyle C=\prod _{k=1}^{\infty }(1-a^{k})}$, avec ${\displaystyle a=-{\frac {1}{\phi ^{2}}}}$ et où ${\displaystyle \phi }$ est encore le nombre d'or.

An approximate truncated value of ${\displaystyle C}$ is 1.226742010720 (see (sequence A062073 in the OEIS) for more digits).

## Almost-Fibonorial numbers

Almost-Fibonorial numbers: n!F − 1.

Almost-Fibonorial primes: prime numbers among the almost-Fibonorial numbers.

## Quasi-Fibonorial numbers

Quasi-Fibonorial numbers: n!F + 1.

Quasi-Fibonorial primes: prime numbers among the quasi-Fibonorial numbers.

## Connection with the q-Factorial

The fibonorial can be expressed in terms of the q-factorial and the golden ratio ${\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}$:

${\displaystyle n!_{F}=\phi ^{\binom {n}{2}}\,[n]_{-1/\phi ^{2}}!}$

## Sequences

Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

and for n such that n!F − 1 and n!F + 1 are primes, respectively.