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.
Read more about this topic: Finite Difference
Famous quotes containing the word difference:
“The real grounds of difference upon important political questions no longer correspond with party lines.... Politics is no longer the topic of this country. Its important questions are settled... Great minds hereafter are to be employed on other matters.... Government no longer has its ancient importance.... The people’s progress, progress of every sort, no longer depends on government. But enough of politics. Henceforth I am out more than ever.”
—Rutherford Birchard Hayes (1822–1893)