Stirling's Finite Difference Formula

$$f_p=f_0+{\textstyle{1\over 2}}p(\delta_{1/2}+\delta_{-1/2})+... ..._0^2+S_3(\delta_{1/2}^2+\delta_{-1/2}^2)+S_4\delta_0^4+\ldots,$$

for $p\in[-1/2,1/2]$, where $\delta$ is the Central Difference and

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

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

with ${n\choose k}$ a Binomial Coefficient.

See also Central Difference, Steffenson's Formula

References

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