Define f to be the Fibonacci sequence:

Calculate the 20th Fibonacci number:

Compare the result with the built-in Fibonacci function:

Plot the first 10 Fibonacci numbers:

Calculate the sum of the first 30 Fibonacci numbers:

Find the generating function for the Fibonacci sequence:

Find the first 10 coefficients of the series expansion of :

Compare the result with the first 10 Fibonacci numbers:

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

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[1]=2 so that it is defined for all n:

Use DifferenceRoot to get the difference equation form of HarmonicNumber:

Reduce combinations of special sequences to their DifferenceRoot forms:

Use f like any sequence:

Define the Pell number sequence using DifferenceRoot:

Use it like any sequence:

Prove properties of Pell numbers:

Closed-form formula:

A summation identity:

Reduce combinations of special sequences to a DifferenceRoot function:

Plot it:

Generate a function for which the Taylor expansion is the given DifferenceRoot object:

Verify the result:

Generate the DifferenceRoot object that generates the BesselJ functions:

Use DifferenceRootReduce to generate DifferenceRoot objects:

Get the corresponding ordinary difference equation:

Use the equation to verify solutions:

Sum of a DifferenceRoot object:

GeneratingFunction may generate a DifferentialRoot object from a holonomic sequence:

For specific cases, GeneratingFunction may give an explicit function:

Find the exponential generating function of a DifferenceRoot object:

The solution of a difference equation may be a DifferenceRoot object:

A result from Sum may be a DifferenceRoot object:

Coefficients in the expansion of a function may be given as a DifferenceRoot object:

The FindSequenceFunction result may be a DifferenceRoot object:

FunctionExpand attempts to generate simpler expressions for DifferenceRoot:

FunctionExpand attempts to generate simpler expressions for parametric sequences:

Define f to be some holonomic sequence:

Compare the result with the output of the RecurrenceTable:

DiscreteShift takes a DifferenceRoot function and generates a shifted sequence:

DifferenceDelta takes a DifferenceRoot function as an input:

Simplify this expression:

DifferenceRoot evaluates only linear difference sequences with polynomial coefficients:

DifferenceRoot evaluates only integer terms:

Define a DifferenceRoot function:

Plot it:

Prove identities:

Attempt to expand into less general functions: