TOPICS
Search

Fibonacci Factorial Constant

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The Fibonacci factorial constant is the constant appearing in the asymptotic growth of the fibonorials (aka. Fibonacci factorials) n!_F. It is given by the infinite product

F=product_(k=1)^infty(1-a^k),
(1)

where

a=-1/(phi^2)
(2)

and phi is the golden ratio.

It can be given in closed form by

F=(-phi^(-2);-phi^(-2))_infty
(3)
=((-1)^(1/24)phi^(1/12))/(2^(1/3))[theta_1^'(0,-i/phi)]^(1/3)
(4)
=1.2267420...
(5)

(OEIS A062073), where (q;q)_infty is a q-Pochhammer symbol and theta_n(z,q) is a Jacobi theta function.

See also

Fibonorial, Golden Ratio, Infinite Product

Explore with Wolfram|Alpha

References

Finch, S. R. "Fibonacci Factorials." §1.2.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 10, 2003.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 478 and 571, 1994.Plouffe, S. http://pi.lacim.uqam.ca/piDATA/fibofact.txt.Sloane, N. J. A. Sequence A062073 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Fibonacci Factorial Constant

Cite this as:

Weisstein, Eric W. "Fibonacci Factorial Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciFactorialConstant.html

Subject classifications