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:
“But that’s always the way; it don’t make no difference whether you do right or wrong, a person’s conscience ain’t got no sense, and just goes for him anyway.... It takes up more room than all the rest of a person’s insides, and yet ain’t no good, nohow. Tom Sawyer thinks the same.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)