Fibonacci Hyperbolic Sine

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

$\displaystyle \mathop{\rm sFh}(x)$	$\textstyle \equiv$	$\displaystyle {\psi^x-\psi^{-x}\over\sqrt{5}}$	(3)
	$\textstyle =$	$\displaystyle {\phi^{2x}-\phi^{-2x}\over\sqrt{5}}$	(4)
	$\textstyle =$	$\displaystyle {2\over\sqrt{5}}\sinh[2x\alpha].$	(5)

For $n\in\Bbb{Z}$ , $\mathop{\rm sFh}(n)=F_{2n}$ where

is a Fibonacci Number. The function satisfies

$\begin{displaymath} \mathop{\rm sFh}(-x)=-\mathop{\rm sFh}(x). \end{displaymath}$

(6)

References

Trzaska, Z. W. ``On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles.'' Fib. Quart. 34, 129-138, 1996.