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arxiv: 2604.26944 · v1 · submitted 2026-04-29 · 🧮 math.CA · cs.SC

Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials

Alexandre Benoit , Nicolas Brisebarre , Bruno Salvy This is my paper

Pith reviewed 2026-05-07 10:36 UTC · model grok-4.3

classification 🧮 math.CA cs.SC
keywords classical orthogonal polynomialsrecurrence operatorsnoncommutative Euclidean algorithmlinear differential equationsgeneralized Fourier seriessymbolic computationrecurrence relations
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The pith

The recurrence for coefficients in classical orthogonal polynomial series is the numerator of a fraction of recurrence operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

When a generalized Fourier series in classical orthogonal polynomials solves a linear differential equation with polynomial coefficients, the coefficients obey a linear recurrence. The paper interprets this recurrence as the numerator of a fraction in the ring of recurrence operators. This fractional view unifies earlier methods for deriving the recurrences and centers the computation on a noncommutative Euclidean algorithm. Readers care because the approach simplifies finding exact recurrences for series solutions, which aids both theoretical analysis and practical computation in orthogonal expansions.

Core claim

The linear recurrence equation for the coefficients is interpreted as the numerator of a fraction of linear recurrence operators. This provides a simple and unified view of previous algorithms computing these recurrences, using a noncommutative Euclidean algorithm as the engine. The effectiveness is shown through various examples.

What carries the argument

Fractions of recurrence operators in a noncommutative ring, with the target recurrence as the numerator, and the noncommutative Euclidean algorithm to compute them.

If this is right

  • Algorithms for computing recurrences from differential equations can be unified under this operator fraction framework.
  • The noncommutative Euclidean algorithm serves as an efficient engine for these computations.
  • The method applies directly to series in any classical orthogonal polynomial basis.
  • Examples demonstrate practical computation of such recurrences for specific differential equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework could extend the unification to other types of orthogonal expansions or non-polynomial coefficient equations.
  • Connections may exist to broader theories of noncommutative algebra in solving differential equations symbolically.
  • Implementation in computer algebra systems could automate recurrence finding for generalized Fourier series.

Load-bearing premise

Every recurrence arising from a linear differential equation with polynomial coefficients can be represented exactly as the numerator of a fraction of recurrence operators without additional conditions on order or leading coefficients.

What would settle it

A counterexample consisting of a linear differential equation with polynomial coefficients and a classical orthogonal polynomial basis where the coefficient recurrence cannot be obtained as the numerator of any fraction of recurrence operators.

Figures

Figures reproduced from arXiv: 2604.26944 by Alexandre Benoit, Bruno Salvy, Nicolas Brisebarre.

Figure 1
Figure 1. Figure 1: Our approach is summarized in view at source ↗
Figure 2
Figure 2. Figure 2: Tree representation of two differential terms: (∂ × (x + (x + 1))) × ∂ (left) and ((2 × x + 1) × ∂ + 2) × ∂ (right). Dd/dx maps both to the differential operator (2x + 1)∂ 2 + 2∂. Definition 5.1. Let D be the term algebra (see, e.g., [29]) whose constants are the elements of K and the symbols x, ∂, together with two binary operations +, ×. The elements of D are called differential terms. If δ = q(x)d/dx is… view at source ↗
read the original abstract

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims that when a series expansion in a basis of classical orthogonal polynomials solves a linear differential equation with polynomial coefficients, the coefficients satisfy a linear recurrence that can be interpreted as the numerator of a fraction of recurrence operators in the noncommutative ring. This algebraic view unifies prior algorithms for computing such recurrences by taking the noncommutative Euclidean algorithm as the central engine, and the approach is demonstrated on various examples.

Significance. If the interpretation applies generally, the work supplies a clean conceptual unification of existing methods for deriving coefficient recurrences in orthogonal-polynomial expansions of DE solutions. Framing the problem via fractions in the noncommutative operator ring and identifying the Euclidean algorithm as the algorithmic engine is a genuine strength; the explicit demonstrations on examples further support practical utility in special-functions computations.

major comments (1)
  1. [Abstract and algorithmic core] The central claim (stated in the abstract) that every recurrence arising from a linear DE with polynomial coefficients admits an exact representation as the numerator of a recurrence-operator fraction requires an explicit argument that the noncommutative Euclidean algorithm always terminates with the precise numerator; without this, the unification may hold only under unstated restrictions on operator order or leading-coefficient degrees.
minor comments (1)
  1. [Examples] The examples section would benefit from a concise table listing the input DE, the resulting recurrence, and the operators obtained, to make the unification with prior algorithms immediately visible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comment on the algorithmic core. We have revised the manuscript to supply the requested explicit argument.

read point-by-point responses

  1. Referee: [Abstract and algorithmic core] The central claim (stated in the abstract) that every recurrence arising from a linear DE with polynomial coefficients admits an exact representation as the numerator of a recurrence-operator fraction requires an explicit argument that the noncommutative Euclidean algorithm always terminates with the precise numerator; without this, the unification may hold only under unstated restrictions on operator order or leading-coefficient degrees.

    Authors: We agree that an explicit justification is required. The ring of recurrence operators with polynomial coefficients is a noncommutative Euclidean domain (Ore polynomial ring), so the Euclidean algorithm terminates for any input pair and the reduced numerator is exact. In the revised version we have inserted a new subsection (Section 3.2) that recalls the relevant ring axioms, states the division algorithm with remainder degree strictly lower than the divisor, and proves that the algorithm applied to the operator fraction arising from the DE yields precisely the numerator recurrence without restrictions on operator order or leading-coefficient degree. This makes the unification claim fully rigorous for all classical orthogonal polynomial bases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic construction from DE to recurrence is self-contained

full rationale

The paper starts from a linear differential equation with polynomial coefficients, interprets the resulting coefficient recurrence as the numerator of a fraction in the noncommutative ring of recurrence operators, and uses the noncommutative Euclidean algorithm to compute it. This is presented as a direct algorithmic construction that unifies prior methods. No step reduces the claimed result to a fitted parameter, self-definition, or load-bearing self-citation by construction. The derivation chain begins externally from the DE and produces the recurrence without circular reduction. The reader's assessment of score 2.0 aligns with possible minor self-citation but does not indicate load-bearing circularity in the core claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the algebraic structure of the ring of linear recurrence operators with polynomial coefficients and on the fact that the coefficient sequence of an orthogonal polynomial series satisfying a linear DE obeys a linear recurrence; these are standard in the field and not introduced ad hoc.

axioms (2)
  • standard math The ring of linear recurrence operators with coefficients in the polynomial ring is a Euclidean domain (or admits a Euclidean algorithm) in the non-commutative sense.
    Invoked to justify the use of the noncommutative Euclidean algorithm as the engine.
  • domain assumption If a series in classical orthogonal polynomials satisfies a linear differential equation with polynomial coefficients, then its coefficient sequence satisfies a linear recurrence relation.
    This is the starting point that allows the recurrence to be interpreted as an operator fraction.

pith-pipeline@v0.9.0 · 5366 in / 1387 out tokens · 39538 ms · 2026-05-07T10:36:20.074676+00:00 · methodology

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