Euler transform

In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.

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

Formally, one may write

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 ${\displaystyle \delta _{nm}}$ is the Kronecker delta. The original series can be regained by

The binomial transform of a sequence is just the nth forward differences 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

In this case the former transform is called the inverse binomial transform, and the latter is just binomial transform. This is standard usage for example in On-Line Encyclopedia of Integer Sequences.

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