Sequence transformation

In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods.

Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others.

For a given sequence

the transformed sequence is

where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.

for some ${\displaystyle k}$ which often depends on ${\displaystyle n}$ (cf. e.g. Binomial transform). In the simplest case, the ${\displaystyle s_{n}}$ and the ${\displaystyle s'_{n}}$ are real or complex numbers. More generally, they may be elements of some vector space or algebra.

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