Central Difference

The central difference for a function tabulated at equal intervals $f_i$ is defined by

$$\delta(f_{n+1/2})=\delta_{n+1/2}=\delta^1_{n+1/2}\equiv f_{n+1}-f_n.$$ (1)

Higher order differences may be computed for Even and Odd powers,

$$\delta_{n+1/2}^{2k} = \sum_{j=0}^{2k} (-1)^j{2k\choose j} f_{n+k-j}$$ (2)

$$\delta_{n+1/2}^{2k+1} = \sum_{j=0}^{2k+1} (-1)^j{2k+1\choose j}f_{n+k+1-j}.$$ (3)

See also Backward Difference, Divided Difference, Forward Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Differences.'' §25.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.