Positivity of arbitrary-order P-recursive sequences with a unique dominant root

Zhongjie Li

arxiv: 2605.17013 · v1 · pith:QERSKWHGnew · submitted 2026-05-16 · 🧮 math.CO

Positivity of arbitrary-order P-recursive sequences with a unique dominant root

Zhongjie Li This is my paper

Pith reviewed 2026-05-19 20:00 UTC · model grok-4.3

classification 🧮 math.CO

keywords P-recursive sequencesultimate positivitydominant rootlinear recurrenceasymptotic signcombinatorial sequencespositivity criteria

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The pith

A sufficient condition proves ultimate positivity for P-recursive sequences of any order with a unique dominant root.

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

The paper shows that P-recursive sequences satisfying a linear recurrence with polynomial coefficients admit a sufficient condition for ultimate positivity whenever they possess one characteristic root strictly larger in modulus than all others. Once this condition holds, checking only finitely many initial terms decides whether every term is positive. The method applies equally to recurrences of order two, three, or higher and is demonstrated on several explicit examples of order greater than two. A reader would care because many combinatorial counts obey such recurrences and their sign patterns control enumeration, generating functions, and asymptotic formulas.

Core claim

We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an application, we provide several examples of P-recursive sequences of order greater than two.

What carries the argument

The unique dominant root of the characteristic equation, which strictly dominates all other roots in modulus and thereby controls the sign of the sequence for all sufficiently large indices.

If this is right

Ultimate positivity of the tail follows directly once the sufficient condition on the dominant root is verified.
Complete positivity of the sequence reduces to a finite check after the sufficient condition holds.
The same criterion works for recurrences of every fixed order, including those greater than two.
Explicit examples confirm that the condition can be checked in practice for concrete sequences.

Where Pith is reading between the lines

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

The criterion may simplify sign proofs for combinatorial sequences such as those counting lattice paths or tilings that obey higher-order recurrences.
It suggests a route to positivity results for sequences whose generating functions satisfy polynomial-coefficient differential equations.
Extensions could examine borderline cases where two roots share the same maximal modulus.

Load-bearing premise

The sequence possesses exactly one dominant root of its characteristic equation.

What would settle it

An explicit P-recursive sequence with a unique dominant root that satisfies the stated sufficient condition yet takes negative values at infinitely many indices.

read the original abstract

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

This paper gives a practical sufficient condition for eventual positivity in P-recursive sequences of any order when there is a unique dominant root.

read the letter

The main point is that this paper supplies a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order, assuming they have a unique dominant root in the characteristic equation. It generalizes earlier work on low-order cases by showing that the dominant root dictates the sign of the terms for all sufficiently large n. Once that is established, positivity reduces to checking a finite initial segment of the sequence. The authors apply this to a few examples of order greater than two, which helps make the method concrete. The reasoning looks straightforward and free of obvious gaps. The asymptotic analysis follows the usual pattern for solutions to linear recurrences, and the finite-check step is presented as a practical way to resolve the question completely. There is no indication that the condition depends on fitting parameters or loops back on itself. The clearest limitation is the unique dominant root requirement. Sequences with multiple roots of comparable size fall outside the result, and the paper does not address how to handle those. This is not a problem with the proof but does restrict the range of sequences it covers directly. This work is aimed at researchers in enumerative combinatorics who encounter P-recursive sequences and need to establish positivity properties. A reader who works with generating functions or recurrence relations for counting problems would find a usable tool here. The paper shows honest engagement with the existing literature on positivity criteria and extends it in a verifiable way. I would bring this to the next reading group to see how it applies to some standard sequences. It deserves a serious referee because the generalization to arbitrary order is substantive and the approach is grounded enough to warrant detailed review. I recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order possessing a unique dominant root of the characteristic equation. Under this hypothesis the asymptotic sign is governed by the contribution of the dominant root, so that ultimate positivity reduces to a finite check of initial terms; several explicit examples of order greater than two are supplied as applications.

Significance. If the arguments are correct, the result supplies a practical, effective criterion for a class of sequences that appear frequently in enumerative combinatorics and computer algebra. The reduction to a finite verification step is particularly useful, and the concrete examples for orders greater than two illustrate applicability beyond the low-order cases already treated in the literature.

minor comments (3)

The statement of the main sufficient condition (presumably Theorem 3.1 or 3.2) would be clearer if the precise algebraic relation between the dominant root and the leading coefficient were written explicitly rather than left implicit in the asymptotic analysis.
In the examples section, the initial terms whose signs are checked should be listed numerically together with the recurrence coefficients so that the finite verification can be reproduced without additional computation.
A short paragraph relating the new criterion to existing positivity results for order-1 and order-2 P-recursive sequences would help readers place the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. We appreciate the recognition that the sufficient condition provides a practical criterion for ultimate positivity in this class of sequences and that the higher-order examples demonstrate broader applicability.

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions identified

full rationale

The paper's sufficient condition for ultimate positivity rests on the explicit hypothesis of a unique dominant root in the characteristic equation of the P-recurrence. Asymptotics are derived directly from the recurrence to show that the dominant term governs the sign for large n, after which finitely many initial terms are checked by direct computation. No equations reduce a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported via self-citation, and the argument does not rename a known empirical pattern. The derivation is therefore independent of its own outputs and remains within standard linear recurrence theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or audited from the given information.

pith-pipeline@v0.9.0 · 5550 in / 1119 out tokens · 50139 ms · 2026-05-19T20:00:01.341361+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Suppose that lim a_n / a_{n-1} = μ. Then there exists u ≥ r such that p < a_n / a_{n-1} < q for n ≥ u (Theorem 2.2).
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Characteristic polynomial p(t) = t^d − Σ ω_j (b_1j^(k)/a_2j^(k)) t^{d-j} with unique positive dominant root μ.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Bell and S

J.P. Bell and S. Gerhold, On the positivity set of a linear recurrence sequence, Israel J. Math.157(2006), 333–345. 13

work page 2006
[2]

Cusick, Recurrences for sums of powers of binomial coefficients, J

T.W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory Ser. A52(1989), 77–83

work page 1989
[3]

Deutsch and L

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math.241(2001), 241–265

work page 2001
[4]

Halava, T

V. Halava, T. Harju and M. Hirvensalo, Positivity of second order linear recurrent sequences, Discrete Appl. Math.154(2006), 447–451

work page 2006
[5]

Ibrahim and B

A. Ibrahim and B. Salvy, Positivity Certificates for Linear Recurrences, Proc. 2024 Annu. ACM-SIAM Symp. Discrete Algorithms (SODA) (2024), 982–994

work page 2024
[6]

Kauers and V

M. Kauers and V. Pillwein, When can we detect that a P-finite sequence is positive?, Proc. 2010 Int. Symp. Symbolic Algebraic Comput. (ISSAC) (2010), 195–201

work page 2010
[7]

Kenison, O

G. Kenison, O. Klurman, E. Lefaucheux, F. Luca, P. Moree, J. Ouak- nine, M. A. Whiteland and J. Worrell, On Positivity and Minimality for Second-Order Holonomic Sequences, Proc. 46th Int. Symp. Math. Found. Comput. Sci. (MFCS), Leibniz Int. Proc. Inform.202(2021), 67:1–67:15

work page 2021
[8]

Laohakosol and P

V. Laohakosol and P. Tangsupphathawat, Positivity of third order linear recurrence sequences, Discrete Appl. Math.157(2009), 3239–3248

work page 2009
[9]

Liu, Positivity of three-term recurrence sequences, Electron

L.L. Liu, Positivity of three-term recurrence sequences, Electron. J. Combin.17(2010), #R57

work page 2010
[10]

R. J. Mcintosh, Recurrences for Alternating Sums of Powers of Binomial Coefficients, J. Combin. Theory Ser. A63(1993), 223–233

work page 1993
[11]

Ouaknine and J

J. Ouaknine and J. Worrell, On the Positivity Problem for Simple Lin- ear Recurrence Sequences, Proc. Int. Colloq. Autom. Lang. Program. (ICALP), Lect. Notes Comput. Sci.8573(2014), 318–329

work page 2014
[12]

Ouaknine and J

J. Ouaknine and J. Worrell, Positivity Problems for Low-Order Linear Recurrence Sequences, Proc. Twenty-Fifth Annu. ACM-SIAM Symp. Discrete Algorithms (SODA) (2014), 366–379. 14

work page 2014
[13]

Ouaknine and J

J. Ouaknine and J. Worrell, Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences, Proc. Int. Colloq. Autom. Lang. Program. (ICALP), Lect. Notes Comput. Sci.8573(2014), 330–341

work page 2014
[14]

Pillwein, On the positivity of the Gillis-Reznick-Zeilberger rational function, Adv

V. Pillwein, On the positivity of the Gillis-Reznick-Zeilberger rational function, Adv. Appl. Math.104(2019), 75–84

work page 2019
[15]

Y. Pei, Y. Wang and Y. Wang, Positivity problem of three-term recur- rence sequences, Linear Algebra Appl.668(2023), 93–109

work page 2023
[16]

Stanley, Positivity problems and conjectures in algebraic combi- natorics, Mathematics: Frontiers and Perspectives, Amer

R.P. Stanley, Positivity problems and conjectures in algebraic combi- natorics, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319

work page 2000
[17]

Straub, Positivity of Szeg¨ o’s rational function, Adv

A. Straub, Positivity of Szeg¨ o’s rational function, Adv. Appl. Math.41 (2008), 255–264

work page 2008
[18]

Xia and X.M

E.X.W. Xia and X.M. Yao, The signs of three-term recurrence sequences, Discrete Appl. Math.159(2011), 2290–2296. 15

work page 2011

[1] [1]

Bell and S

J.P. Bell and S. Gerhold, On the positivity set of a linear recurrence sequence, Israel J. Math.157(2006), 333–345. 13

work page 2006

[2] [2]

Cusick, Recurrences for sums of powers of binomial coefficients, J

T.W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory Ser. A52(1989), 77–83

work page 1989

[3] [3]

Deutsch and L

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math.241(2001), 241–265

work page 2001

[4] [4]

Halava, T

V. Halava, T. Harju and M. Hirvensalo, Positivity of second order linear recurrent sequences, Discrete Appl. Math.154(2006), 447–451

work page 2006

[5] [5]

Ibrahim and B

A. Ibrahim and B. Salvy, Positivity Certificates for Linear Recurrences, Proc. 2024 Annu. ACM-SIAM Symp. Discrete Algorithms (SODA) (2024), 982–994

work page 2024

[6] [6]

Kauers and V

M. Kauers and V. Pillwein, When can we detect that a P-finite sequence is positive?, Proc. 2010 Int. Symp. Symbolic Algebraic Comput. (ISSAC) (2010), 195–201

work page 2010

[7] [7]

Kenison, O

G. Kenison, O. Klurman, E. Lefaucheux, F. Luca, P. Moree, J. Ouak- nine, M. A. Whiteland and J. Worrell, On Positivity and Minimality for Second-Order Holonomic Sequences, Proc. 46th Int. Symp. Math. Found. Comput. Sci. (MFCS), Leibniz Int. Proc. Inform.202(2021), 67:1–67:15

work page 2021

[8] [8]

Laohakosol and P

V. Laohakosol and P. Tangsupphathawat, Positivity of third order linear recurrence sequences, Discrete Appl. Math.157(2009), 3239–3248

work page 2009

[9] [9]

Liu, Positivity of three-term recurrence sequences, Electron

L.L. Liu, Positivity of three-term recurrence sequences, Electron. J. Combin.17(2010), #R57

work page 2010

[10] [10]

R. J. Mcintosh, Recurrences for Alternating Sums of Powers of Binomial Coefficients, J. Combin. Theory Ser. A63(1993), 223–233

work page 1993

[11] [11]

Ouaknine and J

J. Ouaknine and J. Worrell, On the Positivity Problem for Simple Lin- ear Recurrence Sequences, Proc. Int. Colloq. Autom. Lang. Program. (ICALP), Lect. Notes Comput. Sci.8573(2014), 318–329

work page 2014

[12] [12]

Ouaknine and J

J. Ouaknine and J. Worrell, Positivity Problems for Low-Order Linear Recurrence Sequences, Proc. Twenty-Fifth Annu. ACM-SIAM Symp. Discrete Algorithms (SODA) (2014), 366–379. 14

work page 2014

[13] [13]

Ouaknine and J

J. Ouaknine and J. Worrell, Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences, Proc. Int. Colloq. Autom. Lang. Program. (ICALP), Lect. Notes Comput. Sci.8573(2014), 330–341

work page 2014

[14] [14]

Pillwein, On the positivity of the Gillis-Reznick-Zeilberger rational function, Adv

V. Pillwein, On the positivity of the Gillis-Reznick-Zeilberger rational function, Adv. Appl. Math.104(2019), 75–84

work page 2019

[15] [15]

Y. Pei, Y. Wang and Y. Wang, Positivity problem of three-term recur- rence sequences, Linear Algebra Appl.668(2023), 93–109

work page 2023

[16] [16]

Stanley, Positivity problems and conjectures in algebraic combi- natorics, Mathematics: Frontiers and Perspectives, Amer

R.P. Stanley, Positivity problems and conjectures in algebraic combi- natorics, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319

work page 2000

[17] [17]

Straub, Positivity of Szeg¨ o’s rational function, Adv

A. Straub, Positivity of Szeg¨ o’s rational function, Adv. Appl. Math.41 (2008), 255–264

work page 2008

[18] [18]

Xia and X.M

E.X.W. Xia and X.M. Yao, The signs of three-term recurrence sequences, Discrete Appl. Math.159(2011), 2290–2296. 15

work page 2011