# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# 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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Use f like any sequence:

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Define a new sequence using DifferenceRoot directly:

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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