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Bessel's Finite Difference Formula

An Interpolation formula also sometimes known as

\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)

where $G_k$ are the Coefficients from Gauss's Backward Formula and Gauss's Forward Formula and $E_k$ and $F_k$ are the Coefficients from Everett's Formula. The $B_k$s also satisfy
$\displaystyle B_{2n}(p)$ $\textstyle =$ $\displaystyle B_{2n}(q)$ (6)
$\displaystyle B_{2n+1}(p)$ $\textstyle =$ $\displaystyle -B_{2n+1}(q),$ (7)

q\equiv 1-p.
\end{displaymath} (8)

See also Everett's Formula


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

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 90-91, 1990.

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

© 1996-9 Eric W. Weisstein