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.


 * $${n!}_F := \prod_{i=1}^n F_i,\quad n \ge 0,$$

where $F_{i}$ is the $i$th 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 $$\varphi$$: $$n!_F \sim C \frac {\varphi^{n (n+1)/2}} {5^{n/2}}$$.

Here the fibonorial constant (also called the fibonacci factorial constant ) $$C$$ is defined by $$C = \prod_{k=1}^\infty (1-a^k)$$, where $$a=-\frac{1}{\varphi^2}$$ 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: $n!_{F} &minus; 1$.

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 $$\varphi=\frac{1+\sqrt5}2$$:
 * $$n!_F = \varphi^{\binom n 2} \, [n]_{-\varphi^{-2}}!.$$

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

and for $n$ such that $n!_{F} &minus; 1$ and $n!_{F} + 1$ are primes, respectively.