Built-in Mathematica Symbol

DifferenceRoot

DifferenceRoot[lde]
represents a function that solves the linear difference equation specified by lde[a, n].

MORE INFORMATION

DifferenceRoot works like Function.

DifferenceRoot[lde][s] finds the value of the solution to the difference equation at the specific point s.

DifferenceRoot[lde] essentially gives a representation of the solution for a in RSolve[lde[a, n], a, n].

DifferenceRoot is generated by functions such as Sum, RSolve and SeriesCoefficient.

Functions such as Sum, DifferenceDelta and GeneratingFunction can be used on DifferenceRoot objects.

DifferenceRoot[lde][{s₁, s₂, ...}], etc. threads automatically over lists.

DifferenceRootReduce can be used to reduce combinations of DifferenceRoot objects and other functions to a single DifferenceRoot object.

FunctionExpand will attempt to expand DifferenceRoot objects in terms of ordinary special and elementary functions.

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:

Reduce combinations of special sequences to their DifferenceRoot form:

In[1]:=

Out[1]=

Use f like any sequence:

Define a new sequence using DifferenceRoot directly:

In[1]:=

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 threads element-wise over lists:

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 (1)

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

Applications (1)

Define Pell numbers:

Closed form formula:

Identity analogous to Cassini's identity:

A summation identity:

Properties & Relations (3)

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: