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Stirling's Finite Difference Formula

f_p=f_0+{\textstyle{1\over 2}}p(\delta_{1/2}+\delta_{-1/2})+...

for $p\in[-1/2,1/2]$, where $\delta$ is the Central Difference and
$\displaystyle S_{2n+1}$ $\textstyle =$ $\displaystyle {1\over 2}{p+n\choose 2n+1}$  
$\displaystyle S_{2n+2}$ $\textstyle =$ $\displaystyle {p\over 2n+2}{p+n\choose 2n+1},$  

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

See also Central Difference, Steffenson's Formula


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

© 1996-9 Eric W. Weisstein