Английская Википедия:Fibonorial

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

<math>{n!}_F := \prod_{i=1}^n F_i,\quad n \ge 0,</math>

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

The Fibonorial Шаблон:Math is defined analogously to the factorial Шаблон:Math. 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 <math>\varphi</math>: <math>n!_F \sim C \frac {\varphi^{n (n+1)/2}} {5^{n/2}}</math>.

Here the fibonorial constant (also called the fibonacci factorial constant) <math>C</math> is defined by <math>C = \prod_{k=1}^\infty (1-a^k)</math>, where <math>a=-\frac{1}{\varphi^2}</math> and <math>\varphi</math> is the golden ratio.

An approximate truncated value of <math>C</math> is 1.226742010720 (see Шаблон:OEIS for more digits).

Almost-Fibonorial numbers

Almost-Fibonorial numbers: Шаблон:Math.

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

Quasi-Fibonorial numbers

Quasi-Fibonorial numbers: Шаблон:Math.

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 <math>\varphi=\frac{1+\sqrt5}2</math>:

<math>n!_F = \varphi^{\binom n 2} \, [n]_{-\varphi^{-2}}!.</math>

Sequences

Шаблон:OEIS2C Product of first Шаблон:Math nonzero Fibonacci numbers Шаблон:Math.

Шаблон:OEIS2C and Шаблон:OEIS2C for Шаблон:Math such that Шаблон:Math and Шаблон:Math are primes, respectively.

References

• Шаблон:MathWorld