Fibonorial Explained

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

{n!} _{F} : = \prod ^{n} _{i =1} F _{i,} n \ge 0 ,

where is the ^th Fibonacci number, and gives the empty product (defined as the multiplicative identity, i.e. 1).

The Fibonorial is defined analogously to the factorial . 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 (\varphi ^{n})/(5 ^{n/2}) .

Here the fibonorial constant (also called the fibonacci factorial constant^[1])

C is defined by C = \prod ^{inf ty} _{k =1} ( 1- a ^{k)} , where a =- (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: .

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

Quasi-Fibonorial numbers

Quasi-Fibonorial numbers: .

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

Connection with the q-Factorial

\varphi = (1+ \sqrt 5)/(2) :n! _{F} = \varphi ^{\binom} [n] _{- \varphi ^{-2}} !.

Sequences

Product of first nonzero Fibonacci numbers .

and for such that and are primes, respectively.

Notes and References

1. Web site: Fibonacci Factorial Constant. W.. Weisstein, Eric. mathworld.wolfram.com. en. 2018-10-25.