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Forward Difference

The forward difference is a Finite Difference defined by

\Delta f_p\equiv f_{p+1}-f_p.
\end{displaymath} (1)

Higher order differences are obtained by repeated operations of the forward difference operator, so
$\displaystyle \Delta^2 f_p$ $\textstyle =$ $\displaystyle {\Delta_p}^2=\Delta(\Delta_p)=\Delta(f_{p+1}-f_p)$  
  $\textstyle =$ $\displaystyle \Delta_{p+1}-\Delta_p=f_{p+2}-2f_{p+1}+f_p.$ (2)

In general,
\Delta^k_p \equiv \Delta^k f_p \equiv \sum_{m=0}^k (-1)^m {k\choose m} f_{p+k-m},
\end{displaymath} (3)

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

Newton's Forward Difference Formula expresses $f_p$ as the sum of the $n$th forward differences

f_p=f_0+p\Delta_0+{\textstyle{1\over 2!}}p(p+1)\Delta_0^2+{\textstyle{1\over 3!}}p(p+1)(p+2)\Delta_0^3+\ldots
\end{displaymath} (4)

where $\Delta_0^n$ is the first $n$th difference computed from the difference table.

See also Backward Difference, Central Difference, Difference Equation, Divided Difference, Reciprocal Difference


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 877, 1972.

© 1996-9 Eric W. Weisstein