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
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