Fibonacci Hyperbolic Cosine

Let

$\begin{displaymath} \psi\equiv 1+\phi={\textstyle{1\over 2}}(3+\sqrt{5}\,)\approx 2.618034 \end{displaymath}$

(1)

where $\phi$ is the Golden Ratio, and

$\begin{displaymath} \alpha=\ln\phi\approx 0.4812118. \end{displaymath}$

(2)

Then define

$\displaystyle \mathop{\rm cFh}(x)$	$\textstyle \equiv$	$\displaystyle {\psi^{x+1/2}+\psi^{-(x+1/2)}\over\sqrt{5}}$	(3)
	$\textstyle =$	$\displaystyle {\phi^{(2x+1)}+\phi^{-(2x+1)}\over\sqrt{5}}$	(4)
	$\textstyle =$	$\displaystyle {2\over\sqrt{5}} \cosh[(2x+1)\alpha).$	(5)