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Steffenson's Formula

$f_p=f_0+{\textstyle{1\over 2}}p(p+1)\delta_{1/2}-{\textstyle{1\over 2}}(p-1)p\delta_{-1/2}$
$+(S_3+S_4)\delta_{1/2}^3+(S_3-S_4)\delta_{-1/2}^3+\ldots,\quad$ (1)
for $p\in[-{\textstyle{1\over 2}},{\textstyle{1\over 2}}]$, where $\delta$ is the Central Difference and

$\displaystyle S_{2n+1}$ $\textstyle =$ $\displaystyle {1\over 2}{p+n\choose 2n+1}$ (2)
$\displaystyle S_{2n+2}$ $\textstyle =$ $\displaystyle {p\over 2n+2}{p+n\choose 2n+1}$ (3)
$\displaystyle S_{2n+1}-S_{2n+2}$ $\textstyle =$ $\displaystyle {p+n+1\choose 2n+2}$ (4)
$\displaystyle S_{2n+1}-S_{2n+2}$ $\textstyle =$ $\displaystyle -{p+n\choose 2n+2},$ (5)

where ${n\choose k}$ is a Binomial Coefficient.

See also Central Difference, Stirling's Finite Difference Formula


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

© 1996-9 Eric W. Weisstein