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.
Contents
- Asymptotic behaviour
- Almost Fibonorial numbers
- Quasi Fibonorial numbers
- Connection with the q Factorial
- Sequences
- References
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.
Asymptotic behaviour
The series of fibonorials is asymptotic to a function of the golden ratio
where the fibonorial constant
An approximate truncated value of
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
Sequences
A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).
A059709 and A053408 for n such that n!F − 1 and n!F + 1 are primes, respectively.