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Steffenson's Formula

f_p=f_0+1/2p(p+1)delta_(1/2)-1/2(p-1)pdelta_(-1/2) +(S_3+S_4)delta_(1/2)^3+(S_3-S_4)delta_(-1/2)^3+...,
(1)

for p in [-1/2,1/2], where delta is the central difference and

S_(2n+1)=1/2(p+n; 2n+1)
(2)
S_(2n+2)=p/(2n+2)(p+n; 2n+1)
(3)
S_(2n+1)-S_(2n+2)=(p+n+1; 2n+2)
(4)
S_(2n+1)-S_(2n+2)=-(p+n; 2n+2),
(5)

where (n; k) is a binomial coefficient.

See also

Central Difference, Stirling's Finite Difference Formula

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References

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

Referenced on Wolfram|Alpha

Steffenson's Formula

Cite this as:

Weisstein, Eric W. "Steffenson's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteffensonsFormula.html

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