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 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 ${\displaystyle \varphi }$: ${\displaystyle n!_{F}\sim C{\frac {\varphi ^{n(n+1)/2}}{5^{n/2}}}}$.

Here the fibonorial constant (also called the fibonacci factorial constant^[1]) ${\displaystyle C}$ is defined by ${\displaystyle C=\prod _{k=1}^{\infty }(1-a^{k})}$, where ${\displaystyle a=-{\frac {1}{\varphi ^{2}}}}$ and ${\displaystyle \varphi }$ is the golden ratio.

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 ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$:

${\displaystyle n!_{F}=\varphi ^{\binom {n}{2}}\,[n]_{-\varphi ^{-2}}!.}$

Sequences

OEIS: A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

OEIS: A059709 and OEIS: A053408 for n such that n!_F − 1 and n!_F + 1 are primes, respectively.