DifferenceRoot

DifferenceRoot[lde][k]
gives the holonomic sequence , specified by the linear difference equation .

DifferenceRoot[lde]
represents a pure holonomic sequence .

DetailsDetails

  • Mathematical sequence, suitable for both symbolic and numerical manipulation; also known as holonomic sequence and P-recursive sequence.
  • The holonomic sequence defined by a DifferenceRoot function satisfies a holonomic difference equation with polynomial coefficients and initial values .
  • DifferenceRoot can be used like any other mathematical function.
  • FunctionExpand will attempt to convert DifferenceRoot functions in terms of special functions.
  • The sequences representable by DifferenceRoot include a large number of special sequences.
  • DifferenceRootReduce can convert many special sequences to DifferenceRoot sequences.
  • Holonomic sequences are closed under many operations, including:
  • constant multiple, integer power
    sums and products
    discrete convolution
    discrete shift, difference and sum
  • DifferenceRoot is automatically generated by functions such as Sum, RSolve, and SeriesCoefficient.
  • Functions such as Sum, DifferenceDelta, and GeneratingFunction work with DifferenceRoot inputs.
  • DifferenceRoot automatically threads over lists.

ExamplesExamplesopen allclose all

Basic Examples  (3)Basic Examples  (3)

Reduce combinations of special sequences to their DifferenceRoot form:

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Define a new sequence using DifferenceRoot directly:

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Use it like any sequence:

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Prove properties:

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Several functions can produce closed-form answers by using DifferenceRoot functions:

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Introduced in 2008
(7.0)
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