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arxiv: 2605.08444 · v1 · submitted 2026-05-08 · 🧮 math.CO

Recognition: 1 theorem link

· Lean Theorem

A short proof of Mathar's 2013 recurrence conjecture for the Laguerre sequence~A025166

Tong Niu
Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:59 UTC · model grok-4.3

classification 🧮 math.CO
keywords Laguerre polynomialsexponential generating functionsrecurrence relationsordinary differential equationscoefficient extractionP-recursive sequences
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The pith

The conjectured recurrence for the Laguerre sequence holds because its exponential generating function obeys a first-order linear ODE.

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

The paper shows that the sequence defined by a(n) = -n! 2^n L_n(1/2) satisfies the order-two recurrence a(n) + (-4n+3) a(n-1) + 4(n-1)^2 a(n-2) = 0 for all n at least 2. It does so by first writing down an explicit closed form for the exponential generating function F(x) = -exp(-x/(1-2x))/(1-2x) and then verifying that this F satisfies the ODE (1-2x)^2 F'(x) = (1-4x) F(x). Expanding both sides of the ODE as power series and equating the coefficient of x^n / n! immediately produces the three-term relation among a(n), a(n-1), and a(n-2).

Core claim

The exponential generating function F(x) = -exp(-x/(1-2x))/(1-2x) for the sequence satisfies the first-order linear ODE (1-2x)^2 F'(x) = (1-4x) F(x). Substituting the closed form into the ODE (or differentiating directly) and then reading off the coefficient of x^n / n! on both sides of the equation yields the recurrence a(n) + (-4n+3) a(n-1) + 4(n-1)^2 a(n-2) = 0 for n >= 2.

What carries the argument

The first-order linear ODE (1-2x)^2 F'(x) = (1-4x) F(x) satisfied by the exponential generating function of the sequence.

If this is right

  • The recurrence holds for every integer n at least 2.
  • Terms of the sequence can be computed recursively from the first two values without evaluating Laguerre polynomials.
  • The sequence is P-recursive of order two with polynomial coefficients of degree at most two.
  • Repeated differentiation of the ODE produces higher-order relations or differential equations satisfied by F.

Where Pith is reading between the lines

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

  • The same ODE technique may prove recurrences for sequences generated by Laguerre polynomials evaluated at other fixed arguments.
  • The explicit generating function allows extraction of asymptotic growth rates or integral representations for sums involving a(n).
  • This coefficient-matching argument from a first-order ODE could extend to proving P-recurrence for other classical orthogonal polynomial sequences.
  • The relation between the differential equation and the recurrence suggests analogous proofs for sequences whose generating functions satisfy low-order linear ODEs.

Load-bearing premise

The stated closed-form expression is the correct exponential generating function whose coefficients are exactly the sequence defined by the Laguerre polynomials.

What would settle it

A direct computation showing that the series expansion of the given F(x) differs from the sequence values a(n) for some n, or that the ODE fails to hold identically after substitution of the closed form.

read the original abstract

For the OEIS sequence A025166, defined by $a(n) = -n!\,2^{n}\,L_{n}(1/2)$ where $L_{n}$ is the Laguerre polynomial of degree $n$, R.~J.~Mathar contributed in February 2013 the conjectured order-2 P-recursive recurrence \[ a(n) + (-4n+3)\, a(n-1) + 4(n-1)^{2}\, a(n-2) \;=\; 0, \qquad n \ge 2. \] We give a one-page proof. The exponential generating function $F(x) = -\exp\!\big(-x/(1-2x)\big)/(1-2x)$ satisfies the first-order linear ODE $(1-2x)^{2} F'(x) = (1-4x)\, F(x)$, and Mathar's recurrence then falls out by reading off the coefficient of $x^{n}/n!$. Both steps are short. The supplementary archive includes a SymPy script which checks the ODE identically and the recurrence numerically up to $n = 5000$.

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

0 major / 1 minor

Summary. The paper claims to prove Mathar's 2013 conjecture that the sequence a(n) = -n! 2^n L_n(1/2) (OEIS A025166) satisfies the order-2 P-recurrence a(n) + (-4n+3)a(n-1) + 4(n-1)^2 a(n-2) = 0 for n >= 2. The proof identifies the EGF F(x) = -exp(-x/(1-2x))/(1-2x), shows it obeys the linear ODE (1-2x)^2 F'(x) = (1-4x) F(x), and extracts the recurrence by reading off [x^n/n!] coefficients; a SymPy script verifies the ODE symbolically and the recurrence numerically to n=5000.

Significance. If the EGF identification holds, the result supplies an elementary, one-page confirmation of the conjecture using only direct differentiation of an explicit closed form and the standard shift rules for EGFs. The approach is parameter-free and non-circular, with independent computational reproducibility supplied in the supplement. This strengthens the catalog of explicit P-recursions for Laguerre-derived sequences and serves as a concise template for similar coefficient-extraction arguments.

minor comments (1)
  1. Abstract: the phrase 'reading off the coefficient of x^n/n! in the ODE' is slightly informal; a one-sentence reminder of the precise multiplication-by-n and shift rules used for EGFs would make the extraction step fully self-contained for readers unfamiliar with the technique.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the clear summary of our approach, and the recommendation to accept the manuscript. We appreciate the recognition of the proof's brevity, its use of direct differentiation and coefficient extraction, and the independent verification via the supplementary SymPy script.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins with the sequence definition a(n) = -n! 2^n L_n(1/2) and the standard ordinary generating function for Laguerre polynomials, which directly yields the stated EGF F(x) after substitution y=1/2 and z=2x with the sign/factorial adjustment. The ODE is obtained by explicit differentiation of this closed form (verifiable algebraically and by the paper's SymPy script). The recurrence is then extracted by equating coefficients of x^n/n! in the ODE using standard EGF shift rules. None of these steps is self-definitional, fitted, or dependent on self-citations; all rest on external standard results and direct computation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the standard identification of the EGF for the scaled Laguerre sequence and on the algebraic fact that linear ODEs with polynomial coefficients induce P-recursive relations on their Taylor coefficients; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The closed-form expression F(x) = -exp(-x/(1-2x))/(1-2x) is the exponential generating function whose coefficients are a(n).
    Invoked at the start of the proof to link the sequence definition to the ODE.
  • standard math If an EGF satisfies a linear ODE with polynomial coefficients, then its Taylor coefficients satisfy a linear recurrence whose order and coefficients are read off from the ODE.
    Used to obtain the recurrence from the ODE without further derivation.

pith-pipeline@v0.9.0 · 5501 in / 1487 out tokens · 51819 ms · 2026-05-12T00:59:40.778294+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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