gives the holonomic sequence , specified by the linear difference equation lde[h,k].
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.
Examplesopen allclose all
Basic Examples (2)
Compare the result with the built-in Fibonacci function:
Several functions can produce closed-form answers by using DifferenceRoot functions:
Numerical Evaluation (6)
Function Properties (9)
DifferenceRoot works with linear recurrences:
DifferenceRoot transforms recurrences with rational coefficients to ones with polynomial coefficients:
DifferenceRoot works on inhomogeneous equations with polynomial forcing functions:
DifferenceRoot works with multiple initial values:
Obtain the same result using AsymptoticRSolveValue:
DifferenceRoot can take parameters:
If possible, DifferenceRoot reduces to built-in functions:
Special Sequences (3)
Generate a parametric sequence corresponding to ChebyshevT polynomials:
Extract the difference equation that the derivatives of ChebyshevT obey:
Check the equality of the first 10 terms of this sequence with the direct derivatives of ChebyshevT:
Generalizations & Extensions (2)
Reduce combinations of special sequences to their DifferenceRoot forms:
Define the Pell number sequence using DifferenceRoot:
Reduce combinations of special sequences to a DifferenceRoot function:
Generate a function for which the Taylor expansion is the given DifferenceRoot object:
Properties & Relations (14)
Sum of a DifferenceRoot object:
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:
Coefficients in the expansion of a function may be given as a DifferenceRoot object:
FunctionExpand attempts to generate simpler expressions for parametric sequences:
Compare the result with the output of the RecurrenceTable:
Possible Issues (2)
Neat Examples (1)
Define a DifferenceRoot function:
Wolfram Research (2008), DifferenceRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/DifferenceRoot.html (updated 2020).
Wolfram Language. 2008. "DifferenceRoot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/DifferenceRoot.html.
Wolfram Language. (2008). DifferenceRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DifferenceRoot.html