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

$$S_{2n+1} = {1\over 2}{p+n\choose 2n+1}$$ (2)

$$S_{2n+2} = {p\over 2n+2}{p+n\choose 2n+1}$$ (3)

$$S_{2n+1}-S_{2n+2} = {p+n+1\choose 2n+2}$$ (4)

$$S_{2n+1}-S_{2n+2} = -{p+n\choose 2n+2},$$ (5)

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

See also Central Difference, Stirling's Finite Difference Formula

References

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