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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.
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.
The series of fibonorials is asymptotic to a function of the golden ratio : .
Here the fibonorial constant (also called the fibonacci factorial constant[1]) is defined by , where and is the golden ratio.
An approximate truncated value of is 1.226742010720 (see (sequence A062073 in the OEIS) for more digits).
Almost-Fibonorial numbers: n!F − 1.
Almost-Fibonorial primes: prime numbers among the almost-Fibonorial numbers.
Quasi-Fibonorial numbers: n!F + 1.
Quasi-Fibonorial primes: prime numbers among the quasi-Fibonorial numbers.
The fibonorial can be expressed in terms of the q-factorial and the golden ratio :
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.
![{\displaystyle n!_{F}=\varphi ^{\binom {n}{2}}\,[n]_{-\varphi ^{-2}}!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f17702f0dae6f2af1c88188c6b2e16afeaaad87)
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