DifferentialRootReduce

Reduces a function $f(x)$ to its holonomic (D-finite) representation — $\mathrm{DifferentialRoot}$ , a linear ODE with polynomial(x) coefficients plus enough derivatives at one point, applied straight back to $x$ . Covers $e^{ax}$ , $\sin(ax)$ , $\cos(ax)$ , polynomials, a single-pole rational function, $\ln(1{+}x)$ , $\arctan(x)$ , $\mathrm{erf}(x)$ , BesselJ, AiryAi and AiryBi; declines sums, products, and anything else. Provided by @enumeratio/analytic.

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DifferentialRootReduce(f(x), x)the D-finite reduction of

f(x)

, as

\mathrm{DifferentialRoot}(\ldots)(x)

Typed signatures

Type	Package	Applies
`(expression, symbol) -> expression`	`analytic`	always

Domain: Special functions

Details

Expands about $x_0 = 0$ wherever that's an ordinary point of the ODE (every case except BesselJ, whose coefficient of $y''$ vanishes at $x=0$ ); BesselJ expands about $x_0=1$ instead, matching a local wolframscript's own choice there. A single-pole rational function $A/(x{-}a)$ also moves off $a$ itself.
Every reduction's initial conditions come from evaluating $f$ and its derivatives (via compute-engine's own D) at the expansion point directly, so they stay exact wherever compute-engine's own arithmetic is (e.g. AiryAi(0), or $2/\sqrt\pi$ for erf); BesselJ's derivative uses the standard identity $J_\nu'(x) = (J_{\nu-1}(x) - J_{\nu+1}(x))/2$ .
Verified by evaluating the reduction back (via DifferentialRoot's own Taylor-series stepper) against the original function at several points inside the radius of convergence, in both this package's own tests and against a local wolframscript.
Declines a rational function whose denominator isn't a single linear factor, a BesselJ/AiryAi/AiryBi argument that isn't the plain variable, and any sum or product of two of these (no small closure algorithm for D-finite sums is implemented here — see difference-root.ts's own sum rule for the hypergeometric case, which doesn't carry over to a general ODE).