Product of Fibonacci numbers

Question

I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers.

Does there exist a tight approximation for these kind of things?

Answer 1

By the Binet formula,

$$F_n\approx\frac{\phi^n}{\sqrt5}.$$

Then multiplying the $n$ first estimates

$$P_n\approx a\frac{\phi^{n(n+1)/2}}{\sqrt5^n}.$$

By numerical computation, $a\approx 1.22674201072$.

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We can deduce an expression for the geometric average

$$\sqrt[n]P_n\approx\frac{\phi^{(n+1)/2}}{\sqrt5}\approx\frac{F_n}{\sqrt{5\phi}}.$$