arxiv: 2605.03170 · v2 · submitted 2026-05-04 · 🧮 math.CO

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A short proof of Mathar's 2013 recurrence conjecture for the Meixner sequence A214615

Tong Niu

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Pith reviewed 2026-05-08 17:40 UTC · model grok-4.3

classification 🧮 math.CO

keywords Meixner polynomialsexponential generating functionsrecurrence relationsordinary differential equationsP-recursive sequencesOEIS A214615

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The pith

The exponential generating function exp(arctan t) over sqrt(1 plus t squared) satisfies the ODE (1 plus t squared) F prime equals (1 minus t) F and yields Mathar's recurrence for the Meixner sequence at one.

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

This paper supplies a short proof that the sequence a(n) defined by the nth Meixner polynomial evaluated at x equals 1 obeys the recurrence a(n) minus a(n minus 1) plus (n minus 1) squared times a(n minus 2) equals zero for n at least 2. The argument begins with an explicit exponential generating function for the sequence and shows that this function satisfies a first-order linear differential equation. Equating coefficients in the differential equation then produces the recurrence directly. A reader would care because the proof converts a numerical conjecture into a theorem while also furnishing a compact closed-form generating function that encodes all the sequence values at once.

Core claim

The function F(t) equals exp(arctan t) divided by the square root of 1 plus t squared serves as the exponential generating function for a(n) equals M_n(1). This function satisfies the first-order linear ODE (1 plus t squared) times the derivative of F equals (1 minus t) times F. Substituting the power series for F into the ODE and comparing coefficients of t to the n over n factorial produces the relation a(n) minus a(n minus 1) plus (n minus 1) squared a(n minus 2) equals zero.

What carries the argument

The exponential generating function F(t) equals exp(arctan t) over sqrt(1 plus t squared) together with the first-order linear ODE (1 plus t squared) F prime of t equals (1 minus t) F of t, from which the recurrence is obtained by coefficient comparison.

If this is right

The conjectured recurrence holds for every integer n greater than or equal to 2.
Sequence terms can be generated from the two preceding values using only multiplication by (n minus 1) squared and subtraction.
The same generating function encodes all higher-order identities that follow from repeated differentiation of the ODE.
The sequence is P-recursive of order exactly two.

Where Pith is reading between the lines

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

The explicit generating function opens the possibility of extracting an explicit non-recursive formula for a(n) via other analytic methods.
Analogous closed forms might exist for Meixner polynomials evaluated at other fixed points besides one.
The appearance of arctan and the factor sqrt(1 plus t squared) suggests possible trigonometric or geometric interpretations of the sequence.

Load-bearing premise

The given closed-form expression is the exponential generating function for the sequence a(n) equals the nth Meixner polynomial evaluated at one.

What would settle it

Direct numerical extraction of coefficients from the proposed generating function that fails to match the sequence values for any n, or an algebraic verification that the stated function does not satisfy the differential equation identically.

read the original abstract

For the OEIS sequence A214615, defined by $a(n) = M_{n}(1)$ where $M_{n}$ is the $n$-th Meixner polynomial satisfying $M_{n+1}(x) = x\,M_{n}(x) - n^{2}\,M_{n-1}(x)$, R.~J.~Mathar contributed on 6~March 2013 the conjectured order-2 P-recursive recurrence $a(n) - a(n-1) + (n-1)^{2}\,a(n-2) = 0$ for $n \ge 2$. We give a one-page proof. The exponential generating function $F(t) = \exp\!\bigl(\arctan t\bigr)/\sqrt{1+t^{2}}$ satisfies the first-order linear ODE $(1+t^{2})\,F'(t) = (1-t)\,F(t)$, and Mathar's recurrence then falls out by reading off the coefficient of $t^{n}/n!$. Both steps are short. The supplementary archive includes a SymPy script that checks the ODE identically and the recurrence numerically up to $n = 500$.

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.

Desk Editor's Note private letter to a colleague

Short proof settles the recurrence conjecture but leaves the EGF origin unexplained.

read the letter

The colleague should know two things about this paper. First, it delivers a short proof of the 2013 conjecture on the recurrence for a(n) = M_n(1) where M_n are the Meixner polynomials. Second, the proof relies on an exponential generating function that is stated rather than derived from the recurrence. The paper is new because it settles the open conjecture with a one-page argument. It does well by presenting a direct path: the function F(t) = exp(arctan t)/sqrt(1+t^2) satisfies the ODE (1 + t^2) F'(t) = (1 - t) F(t), and equating coefficients in the exponential series immediately gives the recurrence a(n) - a(n-1) + (n-1)^2 a(n-2) = 0 for n >= 2. The supplementary SymPy script verifies the ODE symbolically and checks the recurrence numerically up to n=500. This keeps the main text clean and the verification transparent. The soft spot is the lack of derivation for the EGF itself. The paper begins by stating that this is the EGF for the sequence and then proceeds to the ODE and the recurrence. There are no steps shown connecting the three-term recurrence M_{n+1}(x) = x M_n(x) - n^2 M_{n-1}(x) at x=1 to this closed form. The numerical agreement confirms that the recurrence holds for the computed terms of a(n), but it does not prove that the series expansion of F matches a(n) from the start. If the EGF was found by some other means, such as solving the recurrence or using known formulas for Meixner polynomials, that should be indicated or referenced. This work is for readers who follow specific OEIS sequences or who are interested in extracting recurrences from generating functions and differential equations. It has limited scope and does not introduce new general techniques. The paper shows clear thinking in how it links the ODE to the coefficient recurrence. It deserves a serious referee to verify the details and to see if the EGF step can be made more self-contained. I recommend sending it for peer review. The argument is compact and the conjecture is resolved, so it is worth the time of a referee even if some expansion on the generating function is needed.

Referee Report

1 major / 1 minor

Summary. The paper provides a short proof of Mathar's 2013 conjecture that the sequence a(n) = M_n(1), where the Meixner polynomials satisfy the three-term recurrence M_{n+1}(x) = x M_n(x) - n^2 M_{n-1}(x), obeys the order-2 P-recurrence a(n) - a(n-1) + (n-1)^2 a(n-2) = 0 for n >= 2. It states the EGF F(t) = exp(arctan t)/sqrt(1+t^2), verifies that this F satisfies the first-order linear ODE (1+t^2) F'(t) = (1-t) F(t), and obtains the recurrence by equating coefficients of t^n/n! after multiplying the ODE by the appropriate factor.

Significance. If the EGF identification holds, the manuscript supplies an elegant, one-page analytic proof that converts the recurrence conjecture into a direct consequence of an ODE satisfied by a closed-form generating function. The supplementary SymPy script, which confirms the ODE identically and the recurrence numerically to n=500, is a clear strength that supplies machine-checked algebraic verification and reproducible evidence.

major comments (1)

[Main argument / abstract] The central steps begin from the asserted EGF F(t) = exp(arctan t)/sqrt(1+t^2) for a(n) = M_n(1) (abstract and main text). No derivation is supplied showing that the Taylor coefficients of this F match the sequence defined by the Meixner three-term recurrence at x=1 (with a(0)=a(1)=1). Because the ODE verification and coefficient extraction are valid only once this identification is established, the proof is incomplete without either an explicit derivation of the EGF from the polynomial recurrence or a reference to a prior result that supplies it.

minor comments (1)

The numerical verification up to n=500 is useful for spot-checking but is redundant once a complete proof is in place; a brief sentence explaining its role as supplementary evidence would clarify its purpose.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of the ODE-based proof and the supplementary SymPy verification. We address the major comment below.

read point-by-point responses

Referee: The central steps begin from the asserted EGF F(t) = exp(arctan t)/sqrt(1+t^2) for a(n) = M_n(1) (abstract and main text). No derivation is supplied showing that the Taylor coefficients of this F match the sequence defined by the Meixner three-term recurrence at x=1 (with a(0)=a(1)=1). Because the ODE verification and coefficient extraction are valid only once this identification is established, the proof is incomplete without either an explicit derivation of the EGF from the polynomial recurrence or a reference to a prior result that supplies it.

Authors: We agree that the manuscript as submitted states the EGF without deriving it from the defining three-term recurrence of the Meixner polynomials at x=1 or citing a source. This identification is a necessary prerequisite, and the current presentation is therefore incomplete on this point. In the revised version we will add a concise derivation of the EGF directly from the recurrence (for instance by substituting x=1 into the known generating function for Meixner polynomials or by solving the coefficient recurrence explicitly) or, if a suitable reference is located, include the citation. This addition will be kept brief so that the overall proof remains short while becoming fully self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation from stated EGF to recurrence is direct and non-reductive.

full rationale

The paper states the EGF F(t) = exp(arctan t)/sqrt(1+t^2) for a(n) = M_n(1), verifies that this F satisfies the ODE (1+t^2)F'(t) = (1-t)F(t) by direct (SymPy-checkable) differentiation, and extracts the target recurrence by comparing [t^n/n!] coefficients in the differentiated series. None of the enumerated circularity patterns apply: the recurrence is not used to define or fit F, no self-citation chain justifies the central premise, and no ansatz or uniqueness result is smuggled in. The assumption that F is the EGF is presented as given (with numerical coefficient checks up to n=500), but this does not create a self-referential loop in which the recurrence is recovered from itself by construction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the identification of the closed-form EGF and the algebraic verification that it satisfies the stated ODE; both are presented without derivation in the abstract.

axioms (2)

domain assumption The exponential generating function for a(n) is exactly F(t) = exp(arctan t) / sqrt(1 + t^2)
This identification is asserted in the abstract; its derivation from the Meixner recurrence is not shown.
standard math The coefficient of t^n / n! in the series expansion of F(t) is a(n)
Standard definition of exponential generating function.

pith-pipeline@v0.9.0 · 5512 in / 1374 out tokens · 48765 ms · 2026-05-08T17:40:28.533721+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

10 extracted references · 3 canonical work pages · cited by 1 Pith paper

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Niu,A short proof of Mathar’s 2016 recurrence conjecture for OEIS A176677, manuscript in preparation, 2026

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