gives the holonomic sequence , specified by the linear difference equation .
represents a pure holonomic sequence .
- 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.
Reduce combinations of special sequences to their DifferenceRoot form:
Define a new sequence using DifferenceRoot directly:
Several functions can produce closed-form answers by using DifferenceRoot functions: