# DifferenceRoot

DifferenceRoot[lde][k]

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

DifferenceRoot[lde]

represents a pure holonomic sequence .

# Details • 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.

# Examples

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## Basic Examples(3)

Reduce combinations of special sequences to their DifferenceRoot form:

Use f like any sequence:

Define a new sequence using DifferenceRoot directly:

Use it like any sequence:

Prove properties:

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

## Scope(10)

Evaluate at an arbitrary point:

DifferenceRoot works on rational coefficients:

Homogeneous linear recurrences:

Inhomogeneous linear recurrences:

Multiple initial values:

Solutions of a difference equation:

A result from Sum:

Coefficients in the expansion of a function:

Formula of a sequence:

## Generalizations & Extensions(2)

Equations with holonomic constant terms are automatically lifted to polynomial coefficients:

The following function is not defined for n>0:

Add the initial value y=2 so that it is defined for all n:

## Applications(1)

Define Pell numbers:

Closed form formula:

Identity analogous to Cassini's identity:

A summation identity:

## Properties & Relations(4)

Use DifferenceRootReduce to generate DifferenceRoot objects:

Get the corresponding ordinary difference equation:

Use the equation to verify solutions:

Sum of a DifferenceRoot object:

Find the generating function of a DifferenceRoot object:

Find the exponential generating function of a DifferenceRoot object:

Introduced in 2008
(7.0)