Fibonorial

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

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

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

The Fibonorial Template:Math is defined analogously to the factorial Template: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 $\phi$: $n{!}_F\sim C\frac{{\phi }^{n(n+1)/2}}{5^{n/2}}$.

Here the fibonorial constant (also called the fibonacci factorial constant) $C$ is defined by $C=\prod _{k=1}^{\mathrm{\infty }}(1-a^k)$, where $a=-\frac{1}{{\phi }^2}$ and $\phi$ is the golden ratio.

An approximate truncated value of $C$ is 1.226742010720 (see Template:OEIS for more digits).

Almost-Fibonorial numbers

Almost-Fibonorial numbers: Template:Math.

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

Quasi-Fibonorial numbers

Quasi-Fibonorial numbers: Template: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 $\phi =\frac{1+\sqrt{5}}{2}$:

$n{!}_F={\phi }^{(\genfrac{}{}{0ex}{}{n}{2})}[n{]}_{-{\phi }^{-2}}!.$

Sequences

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

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

References

• Template:MathWorld