Binomial Transform

The binomial transform, T, of a sequence, {a_n}, is the sequence {s_n} defined by

Formally, one may write (Ta)_n = s_n for the transformation, where T is an infinite-dimensional operator with matrix elements T_nk:

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