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?
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5$\begingroup$ Better start at $F_1$, not $F_0=0$... $\endgroup$– lhfJun 13, 2016 at 19:19
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1 Answer
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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, $a\approx 1.22674201072$.
We can deduce an expression for the geometric average
$$\sqrt[n]P_n\approx\frac{\phi^{(n+1)/2}}{\sqrt5}\approx\frac{F_n}{\sqrt{5\phi}}.$$
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6$\begingroup$ See mathworld.wolfram.com/Fibonorial.html and mathworld.wolfram.com/FibonacciFactorialConstant.html. $\endgroup$– lhfJun 13, 2016 at 19:43
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$\begingroup$ @lhf: glad to know that I equaled Wolfram. $\endgroup$– user65203Jun 13, 2016 at 19:51
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