Markoff's Formulas

Formulas obtained from differentiating Newton's Forward Difference Formula,

$$f'(a_0+ph)={1\over h}\left[{\Delta_0+{\textstyle{1\over 2}}(... ...lta_0^3+\ldots+{d\over dp}{p\choose n}\Delta_0^n}\right]+R_n',$$ (1)

where

$$R_n'=h^nf^{(n+1)}(\xi){d\over dp}{p\choose n+1}+h^{n+1}{p\choose n+1}{d\over dx}f^{(n+1)}(\xi),$$ (2)

${n\choose k}$ is a Binomial Coefficient, and $a_0<\xi<a_n$. Abramowitz and Stegun (1972) and Beyer (1987) give derivatives $h^nf_0^{(n)}$ in terms of $\Delta^k$ and derivatives in terms of $\delta^k$ and $\nabla^k$.

See also Finite Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 883, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 449-450, 1987.