Bessel's Finite Difference Formula

An Interpolation formula also sometimes known as

$\begin{displaymath} f_p=f_0+p\delta_{1/2}+B_2(\delta_0^2+\delta_1^2)+B_3\delta_{1/2}^3+B_4(\delta_0^4+\delta_1^4)+B_5\delta_{1/2}^5+\ldots, \end{displaymath}$

(1)

for $p\in[0,1]$ , where $\delta$ is the Central Difference and

$\displaystyle B_{2n}$	$\textstyle \equiv$	$\displaystyle {\textstyle{1\over 2}}G_{2n}\equiv {\textstyle{1\over 2}}(E_{2n}+F_{2n})$	(2)
$\displaystyle B_{2n+1}$	$\textstyle \equiv$	$\displaystyle G_{2n+1}-{\textstyle{1\over 2}}G_{2n}\equiv{\textstyle{1\over 2}}(F_{2n}-E_{2n})$	(3)
$\displaystyle E_{2n}$	$\textstyle \equiv$	$\displaystyle G_{2n}-G_{2n+1}\equiv B_{2n}-B_{2n+1}$	(4)
$\displaystyle F_{2n}$	$\textstyle \equiv$	$\displaystyle G_{2n+1}\equiv B_{2n}+B_{2n+1},$	(5)