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.

Asymptotic behaviour

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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

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Almost-Fibonorial numbers: n!F − 1.

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

Quasi-Fibonorial numbers

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Quasi-Fibonorial numbers: n!F + 1.

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

Connection with the q-Factorial

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The fibonorial can be expressed in terms of the q-factorial and the golden ratio  :

 

Sequences

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OEISA003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

OEISA059709 and OEISA053408 for n such that n!F − 1 and n!F + 1 are primes, respectively.

References

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  1. ^ W., Weisstein, Eric. "Fibonacci Factorial Constant". mathworld.wolfram.com. Retrieved 2018-10-25.{{cite web}}: CS1 maint: multiple names: authors list (link)