Finite Difference - n-th Difference

n-th Difference

The nth forward difference of a function f(x) is given by

where is the binomial coefficient. Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties.

Forward differences may be evaluated using the Nörlund–Rice integral. The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.