# Integral representation for Fibonacci's numbers

We know that, for example, the Gamma function is a perfect integral representation for the factorial $n!$ for a natural number $n$.

$$\Gamma[n] = \int_0^{+\infty} t^{n-1}e^{-t}\text{d}t = (n-1)!$$

Is there a similar integral representation through which I might find Fibonacci's numbers? Something like

$$F_n = \int_0^{+\infty} F(x, n)\ \text{d}x$$

to obtain the $n-$th Fibonacci's number?

P.s. Not necessarily an integration from zero to infinity.

• No need for integrals. See en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers Dec 1 '15 at 0:57
• Of course, one can always use the Cauchy integral formula on the generating function for the Fibonacci numbers. May 14 '16 at 3:04

One example can be found on the Wolfram functions site $$F_{2n}=\frac n2 \left(\frac32\right)^{n-1}\int_0^{\pi} \left(1+\frac{\sqrt 5}{3}\cos x\right)^{n-1} \sin x \,dx,$$ and another one in this note: $$F_n=\frac1{\sqrt5}\left(\frac{\sqrt5 +1}{2}\right)^n-\frac2\pi \int_0^{\infty}\frac{\sin\frac{x}2}{x}\frac{\cos n x-2\sin x\sin nx}{5\sin^2 x+\cos^2 x}dx.$$