The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined by
Formally, one may write (Ta)n = sn for the transformation, where T is an infinite-dimensional operator with matrix elements Tnk:
The transform is an involution, that is,
or, using index notation,
where δ is the Kronecker delta function. The original series can be regained by
The binomial transform of a sequence is just the nth forward difference of the sequence, with odd differences carrying a negative sign, namely:
where Δ is the forward difference operator.
Some authors define the binomial transform with an extra sign, so that it is not self-inverse:
whose inverse is
Read more about Binomial Transform: Example, Shift States, Ordinary Generating Function, Euler Transform, Exponential Generating Function, Integral Representation, Generalizations
Famous quotes containing the word transform:
“Bees plunder the flowers here and there, but afterward they make of them honey, which is all theirs; it is no longer thyme or marjoram. Even so with the pieces borrowed from others; one will transform and blend them to make a work that is all one’s own, that is, one’s judgement. Education, work, and study aim only at forming this.”
—Michel de Montaigne (1533–1592)