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Everett's Formula

\end{displaymath} (1)

for $p\in[0,1]$, where $\delta$ is the Central Difference and
$\displaystyle E_{2n}$ $\textstyle \equiv$ $\displaystyle G_{2n}-G_{2n+1}\equiv B_{2n}-B_{2n+1}$ (2)
$\displaystyle F_{2n}$ $\textstyle \equiv$ $\displaystyle G_{2n+1}\equiv B_{2n}+B_{2n+1},$ (3)

where $G_k$ are the Coefficients from Gauss's Backward Formula and Gauss's Forward Formula and $B_k$ are the Coefficients from Bessel's Finite Difference Formula. The $E_k$s and $F_k$s also satisfy
$\displaystyle E_{2n}(p)$ $\textstyle =$ $\displaystyle F_{2n}(q)$ (4)
$\displaystyle F_{2n}(p)$ $\textstyle =$ $\displaystyle E_{2n}(q),$ (5)

q\equiv 1-p.
\end{displaymath} (6)

See also Bessel's Finite Difference Formula


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 880-881, 1972.

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 92-93, 1990.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987.

© 1996-9 Eric W. Weisstein