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Zeros of GKP sequences of polynomials

T0 review · 0 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Under a simple sign condition, every GKP polynomial has only real simple zeros that interlace and stay between the two fixed roots of the quadratic factor.

desk verdict Solid, self-contained real-rootedness and extreme-zero asymptotics for a natural recurrence class that unifies several classical families; the sign condition is essential and clearly marked. read the letter →

arxiv 2607.06578 v1 pith:T7OZIIGB submitted 2026-07-02 math.GM

Antonio J. Durán , Mario Pérez , Juan L. Varona This is my paper · ORCID
classification math.GM MSC 33C4526C1030C15
keywords GKPpolynomialsrealzerosinterlacingtangentsecantEulerianJacobiasymptoticsof
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces GKP sequences of polynomials by the first-order recurrence that multiplies the previous derivative by a fixed quadratic with two distinct real roots and adds a linear term whose coefficients form two free sequences. Classic families—tangent and secant polynomials, Eulerian polynomials, and Jacobi polynomials—all arise this way. The central claim is that a uniform inequality on those free coefficients forces every polynomial in the sequence to have only real, simple zeros lying strictly between the two roots of the quadratic, and that consecutive polynomials interlace. When one coefficient sequence is constant and the other eventually constant, the same machinery also yields precise asymptotics for the extreme zeros and shows that finite linear combinations remain real-rooted for large degree, with a controlled number of zeros that escape the original interval.

What carries the argument

The differential operator Ω(a,b)p = (1-x^{2})p' + (a+bx)p together with the sign-change lemmas that locate an odd number of zeros of Ω(a,b)p in every interval determined by consecutive zeros of p and the points ±1. Induction on these lemmas yields global real-rootedness and interlacing.

What would settle it

Compute the first twenty polynomials of a GKP sequence that violates |φ_n| + ψ_n < 0 for some n (or for which the auxiliary polynomial P vanishes at a critical point u±v-2l) and check whether any non-real zeros appear; the paper itself exhibits permanent complex zeros in the latter case.

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Extended reading notes

Core claim

If |φ_n| + ψ_n < 0 for every n, then every GKP polynomial s^{φ,ψ}_n has only simple zeros inside (-1,1) and the zeros of s^{φ,ψ}_{n+1} interlace those of s^{φ,ψ}_n. The same conclusion, after an affine change of variable, holds for the general GKP recurrence built from any quadratic with two distinct real roots.

Load-bearing premise

The uniform sign condition |φ_n| + ψ_n < 0 must hold for every index n; if it fails even for one n the forcing of all zeros into (-1,1) and the interlacing argument can break.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper defines GKP sequences of polynomials via the recurrence p_n = (ax^{2}+bx+c)p'_{n-1} + (φ_n + ψ_n x)p_{n-1} (with ax^{2}+bx+c having two distinct real zeros) and studies their zeros. After an affine reduction to the model family s^{φ,ψ}_n on (−1,1), Theorem 1.1 asserts that the uniform condition |φ_n| + ψ_n < 0 forces all zeros of s_n to be real and simple in (−1,1) with interlacing between consecutive degrees. When ψ is constant the coefficients are symmetric in the φ-parameters, yielding monotonicity of zeros (Theorem 4.2). For constant and eventually-constant parameters the extreme zeros admit explicit asymptotics (Theorems 1.2 and 6.2), obtained from a general coefficient-asymptotics theorem (Theorem 5.2) proved in the appendix; linear combinations of the constant-parameter family are treated in Theorem 1.3, which locates the zeros relative to (−1,0) according to the sign pattern of an auxiliary polynomial P.

Significance. The work supplies a uniform real-rootedness and interlacing theory that covers several classical families (tangent/secant, Eulerian, Jacobi) as special cases of a single recurrence. The elementary but carefully case-split zero lemmas (3.1–3.4) and the general asymptotic engine of Theorem 5.2 are reusable tools; the latter is of independent interest for any coefficient sequence that is asymptotically of the form F(j)G(j)^n with log-concave F and G. The results are self-contained, free of fitted parameters, and the necessity of the sign hypothesis is documented by partial results (Corollary 4.4) and counter-examples (Lemma 6.4).

minor comments (5)
  1. In the statement of Theorem 1.1 the condition is written “|φ_n| + ψ_n < 0 for every n ≥ 0”; the sequences begin at n = 1, so the range should be n ≥ 1 (or the sequences should be extended by a dummy index).
  2. Definition 3.1 of interlacing is non-standard in requiring min(U) < min(V); a brief remark that this is the convention used throughout would prevent confusion with the usual symmetric notion.
  3. The generating function displayed at the end of §5 is stated without proof; a one-line verification from the PDE, or a reference, would be helpful.
  4. In Lemma 5.1 the binomial coefficient (−u + j − 1 choose j) is written with a negative upper index; an explicit remark that it is understood via the usual generalized binomial formula would remove any ambiguity.
  5. A few typographical inconsistencies appear (e.g., “s ϕ,ψ n ” versus “s^{φ,ψ}_n”, occasional missing spaces around operators). A light copy-edit pass would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: zeros and asymptotics are derived from the recurrence and elementary real-analysis lemmas under an explicit sign hypothesis.

full rationale

The paper defines GKP polynomials by the first-order recurrence (1.1)/(1.4) and proves real-rootedness, simplicity and interlacing (Theorem 1.1) by induction from the elementary sign-change lemmas 3.1–3.4 that use only the uniform condition |φ_n|+ψ_n<0. The constant and eventually-constant cases reduce to the same recurrence via the operator identity (4.2)–(4.4) and the linear combination (6.1); their extreme-zero asymptotics follow from the general coefficient-asymptotics theorem 5.2 (proved in the Appendix from log-concavity of F and G) after the explicit coefficient limits of Lemma 5.1 are verified by direct computation. Self-citations supply only background identities (Stirling numbers, generating functions, earlier special cases) that are independently checkable and are not used to force the main conclusions. There are no fitted parameters, no self-referential normalizations, and no reduction of a claimed prediction to an input by construction. The derivation is therefore self-contained against its stated hypotheses.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper is pure mathematics. Load-bearing ingredients are the recurrence definition, the two-distinct-real-zeros hypothesis on the driving quadratic, the sign condition that forces real-rootedness, and standard real-analysis facts about sign changes and log-concave sequences. No numerical free parameters are fitted. The only ‘invented’ object is the GKP class itself, introduced as the object of study and shown to contain classical families.

assumptions (5)
  • domain assumption The quadratic ax²+bx+c has two distinct real zeros (Definition 1.1 / Remark 2.1).
    Used throughout to reduce to the model factor (1−x²) by affine change of variable; without it the interval containing the zeros is undefined.
  • domain assumption ψ_i ≠ −a(i−1) for all i (so that deg p_n = n).
    Stated in Definition 1.1; guarantees the leading coefficient is nonzero.
  • domain assumption |φ_n|+ψ_n < 0 for every n (or |v|+u < 0 in the constant case).
    Hypothesis of Theorem 1.1 and all subsequent real-rootedness statements; enters the sign computations in Lemmas 3.2–3.4.
  • standard math Standard facts on sign changes of polynomials, Rolle-type interlacing, and log-concave sequences (Lemmas 3.1–3.4, 7.1).
    Elementary real analysis used without further proof.
  • domain assumption P(u+v−2l) and P(−u−v+2l) nonzero for all natural l (condition (1.9)/(6.10)).
    Needed for the asymptotic location of escaped zeros in the linear-combination case; when it fails, Lemma 6.4 exhibits permanent complex zeros.
invented entities (1)
  • GKP sequences of polynomials independent evidence
    purpose: Unify tangent, secant, Eulerian, Jacobi and related combinatorial polynomials under a single two-parameter recurrence and study their zeros.
    Defined in Definition 1.1; shown by direct verification to recover the classical families in Section 2. Independent evidence is the reduction to known named polynomials.

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Pith. "Pith review of Zeros of GKP sequences of polynomials." pith.science (2026). https://pith.science/paper/T7OZIIGB

@misc{pith2026260706578,
  author       = {Pith},
  title        = {Pith review of: Zeros of GKP sequences of polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7OZIIGB}},
  note         = {Machine review of arXiv:2607.06578}
}
abstract

Given two sequences $\phi=(\phi_i)_{i\ge 1}$ and $\psi=(\psi_i)_{i\ge 1}$ and numbers $a,b,c$, we introduce the GKP sequence of polynomials $(p_n)_n$ using the following recurrence formula: $p_0 = 1$ and for $n\ge 1$ \[ p_{n}(x) = (ax^2+bx+c) p_{n-1}'(x) + (\phi_{n} + \psi_{n} x)p_{n-1}(x), \] where we assume that $ax^2+bx+c$ has two different real zeros. Tangent, Secant, Eulerian or Jacobi polynomials are examples of GKP sequences of polynomials. In this paper, under mild assumptions we prove that the zeros of the polynomials $p_n$ are real, simple and live between the zeros of $ax^2+bx+c$. Moreover, the zeros of $p_{n+1}$ interlace the zeros of $p_n$. We study in detail the cases when $\psi$ is constant, and $\phi=(\phi_i)_{i\ge 1}$ is constant for $i$ big enough, proving, among other results, asymptotics for the leftmost and rightmost zeros of $p_n$.

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