TITLE: Continuous Fibonacci number F(n)
QUESTION [8 upvotes]: How is a continuous Fibonacci number $F_n$ defined? (like the Gamma function for factorials)
Tried with Binet's formula (Wiki) , but to no luck.
F[n_] = N[GoldenRatio^n - 1/GoldenRatio^n]/Sqrt[5]
Plot[F[n], {n, 3., 12.}, GridLines -> Automatic]
FibonRatio[n_] = N[Fibonacci[n + 1]/Fibonacci[n]];
Plot[{FibonRatio[n], N[GoldenRatio], 1.7}, {n, 3., 12.}, 
 GridLines -> Automatic]

Also using this continuous function definition how is it proved that
$$\lim _{n\rightarrow \infty}\dfrac{F_{n+1}}{F_n} = \phi$$
( where $\phi$ is the GoldenRatio) ?
Appreciate your comments.

REPLY [3 votes]: Since you are using Mathematica, judging by the code in the OP, you can easily obtain the actual expression used by Mathematica for evaluating Fibonacci numbers of non-integer index:
FunctionExpand[Fibonacci[n]]
   ((1/2 (1 + Sqrt[5]))^n - (2/(1 + Sqrt[5]))^n Cos[n π])/Sqrt[5]

In more conventional notation ($\phi$ is the golden ratio, as usual):
$$F_n=\frac{\phi ^n-\phi ^{-n} \cos (\pi  n)}{\sqrt{5}}$$
(This is also the expression given in the Wolfram Functions site and CostaZach's answer.)
This of course doesn't imply that this is the only way to define the Fibonacci numbers for non-integer index. For instance, this function uses
$$f_n=\frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n\sqrt{5}}$$
but unlike the previous formula, this becomes complex-valued in general for real non-integer argument.