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

Here the fibonorial constant (also called the fibonacci factorial constant[1]) $C$ 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 (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 $\varphi ={\frac {1+{\sqrt {5}}}{2}}$ :

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.

References

W., Weisstein, Eric. "Fibonacci Factorial Constant". mathworld.wolfram.com. Retrieved 2018-10-25.

Weisstein, Eric W. "Fibonorial". MathWorld.

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[1] W., Weisstein, Eric. "Fibonacci Factorial Constant". mathworld.wolfram.com. Retrieved 2018-10-25.