DifferenceRoot

A holonomic (P-recursive) sequence object — a linear recurrence with polynomial(n) coefficients plus enough initial values, exactly as DifferenceRootReduce emits it. Applied to a concrete non-negative integer $n_0$, it runs the recurrence forward in exact rational arithmetic; applied to a symbolic $n$ (or anything else it can't run), it stays inert. Provided by @enumeratio/analytic.

Wolfram LanguageDifferenceRoot↗

DifferenceRoot(fn)(n0)the sequence's value at $n_0$, running the recurrence stored in fn forward from whichever initial condition is closest below $n_0$.

Typed signatures

Type	Package	Applies
(any) -> any	analytic	always

Domain: Special functions

Details

• Ordinarily built by DifferenceRootReduce rather than by hand — the examples below construct one directly to show the evaluation contract on its own.
• Evaluation is attached to the native Apply operator (never redeclares it — see derivatives.ts's own Derivative/D hook for the same pattern), so it only intercepts a call whose head is DifferenceRoot(...); every other Apply call is untouched.
• Declines (stays symbolic) at a step whose leading coefficient evaluates to exactly 0 with no initial condition to jump to instead, and at a negative or non-integer target.

Examples

1 more kept as data, checked against the oracles

Basic #

$y(n{+}1) = y(n) + 1$, $y(0){=}0$: $y(5) = 5$

live ↗

In$\mathrm{DifferenceRoot}((y, n)↦\begin{cases}-y(n)+y(n+1)-1=0\\y(0)=0\end{cases})\lhd5$Out$5$

Scope #

Runs the recurrence forward from its anchors: $F_{30}$ in exact arithmetic

live ↗

In$\mathrm{DifferenceRoot}((y, n)↦\begin{cases}-y(n)-y(n+1)+y(n+2)=0\\y(0)=0\\y(1)=1\end{cases})\lhd30$Out$832\,040$

Properties #

A symbolic index leaves the call unevaluated rather than dropping the argument

live ↗

In$\mathrm{DifferenceRoot}((y, n)↦\begin{cases}-y(n)+y(n+1)-1=0\\y(0)=0\end{cases})\lhd n$Out$\mathrm{DifferenceRoot}((y, n)↦\begin{cases}-y(n)+y(n+1)-1=0\\y(0)=0\end{cases})\lhd n$

About

Examples	4
Crosswalk	Wolfram Language

See also: DifferenceRootReduce, DifferentialRoot