Zeros of linear combinations of Hermite polynomials

Antonio J. Dur\'an

REVIEW 4 major objections 5 minor 2 cited by

Zeros of linear combinations of Hermite polynomials

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Pith's one-line read The auxiliary polynomial $P(x)=\sum_{j=0}^K\gamma_j x^{K-j}$ controls real-rootedness of all consecutive Hermite combinations, with exact zero counts in the Appell normalization.

desk verdict Genuinely new results on zeros of Hermite combinations, but the Appell half leans on an unproved imported asymptotic and a couple of terse induction steps; worth a serious referee. read the letter →

arxiv 2505.15330 v1 pith:AF3PIE7U submitted 2025-05-21 math.CA

Antonio J. Durán This is my paper · ORCID

classification math.CA MSC 42C0526C1033C45

keywords zerosofpolynomialsHermiteAppellreal-rootednessinterlacingbackwardshiftoperatormultipleBrenke-typeasymptotics

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 studies polynomials $q_n(x)=\sum_{j=0}^K\gamma_j\tilde H_{n-j}(x)$ formed from $K+1$ consecutive Hermite polynomials, in two normalizations: the standard one ($\tilde H_n=H_n$) and the one making $q_n$ an Appell sequence ($\tilde H_n=H_n/(2^nn!)$, so $q_n'=q_{n-1}$). Its central claim is that in both cases the real-rootedness of every $q_n$ is controlled by the single finite polynomial $P(x)=\sum_{j=0}^K\gamma_jx^{K-j}$. If $P$ has only real zeros, then in both normalizations all $q_n$ are real-rooted with simple zeros, and the zeros of consecutive polynomials alternate. If $P$ has $N_{nr}$ non-real zeros, then in the Appell normalization $q_n$ has exactly $n-N_{nr}$ real zeros and $N_{nr}$ non-real zeros once $n$ is large enough, and in the standard normalization a threshold depending only on those non-real zeros guarantees the same real-rootedness. This reduces an infinite family of questions to one finite polynomial, and it makes the dependence on normalization explicit.

What carries the argument

The load-bearing mechanism is the backward shift $\Lambda f(x)=2xf(x)-f'(x)$, which satisfies $\Lambda H_n=H_{n+1}$ and is a real-zero-increasing operator, together with the auxiliary polynomial $P(x)=\sum_{j=0}^K\gamma_j x^{K-j}$, whose zeros are the single finite source of all real-rootedness information. The paper encodes $q_n$ in a family of generalized Hermite polynomials $h_{\varphi,\psi}$ built from $\Lambda$; when $\psi_i=0$ these generalize multiple Hermite polynomials and their interlacing properties are proved directly. For the Appell normalization the engine is the generating-function limit $$\lim_n \left(\frac{z}{n+1}\right)^n n!\,q_n\!\left(\frac{n+1}{z}\right)=z^K $e^{{-z^2/4}}$P(1/z),$$ uniform on compact sets, which transfers non-real zeros of $P$ into non-real zeros of $q_n$ and gives the exact count in Theorem 1.3.

What would settle it

For the Appell normalization with $K=2$, $\gamma_0=1$, $\gamma_1=0$, $\gamma_2=1$, so $P(x)=x^2+1$ has two non-real zeros, Theorem 1.3 predicts that once $q_n=\tilde H_n+\tilde H_{n-2}$ first has exactly two non-real zeros, all later $q_n$ also have exactly two non-real zeros and $n-2$ real simple zeros. Computing the zeros for $n=3,4,\ldots,100$ and checking that the count stabilizes at two non-real zeros would settle the claim; any later $n$ with a different count is a counterexample.

Watch

Extended reading notes

Core claim

The paper establishes Theorem 1.2 and Theorem 1.3. In the standard normalization, whenever $P(x)=\sum_{j=0}^K\gamma_j x^{K-j}$ has only real zeros, every $q_n$ with $n\ge K$ has only real and simple zeros, and the zeros of $q_{n+1}$ strictly interlace those of $q_n$; if $P$ has non-real zeros, there is a positive integer $n_0$, depending only on $K$ and on the non-real zeros of $P$, such that the same is true for all $n\ge n_0$. In the Appell normalization, the description is exact: $q_n$ is real-rooted for all $n\ge 0$ if and only if $P$ has no non-real zeros, and if $N_{nr}>0$ there is a smallest $n_0$ such that $q_{n_0}$ has exactly $N_{nr}$ non-real zeros, with $q_n$ having exactly $n-N_{nr}$ real zeros and $N_{nr}$ non-real zeros precisely for $n\ge n_0$; in that range all zeros are simple and the real zeros of consecutive polynomials interlace.

Load-bearing premise

The exact real/non-real zero count in the Appell normalization depends on the uniform Brenke-type asymptotic imported from a companion preprint; if that asymptotic fails to be uniform, the conclusion that non-real zeros of $P$ reappear as exactly $N_{nr}$ non-real zeros of $q_n$ does not follow.

Editorial extensions

If this is right

In the standard normalization, $P$ having only real zeros implies that all $q_n$ with $n\ge K$ are real-rooted, simple, and interlaced degree-by-degree.
In the standard normalization with non-real zeros of $P$, the threshold $n_0$ depends only on $K$ and on those non-real zeros, so a single finite check certifies the whole family.
In the Appell normalization, the deficit in real zeros is constant for large $n$: $q_n$ has exactly $n-N_{nr}$ real zeros and $N_{nr}$ non-real zeros.
Both normalizations give a Turan-type inequality $q_{n-1}(x)^2-q_n(x)q_{n-2}(x)>0$ for all real $x$, from a threshold onwards (or for all $n\ge 2$ when all zeros of $P$ are real in the Appell case).
Rescaled zeros have explicit limits: central real zeros follow the trigonometric Mehler-Heine scaling, the extreme real and non-real zeros approach the corresponding zeros of $P$, and the counting measure converges to the semicircle law.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the proof uses only the backward-shift identity and a Brenke-type generating function, the same 'control polynomial $P$' structure should hold for other Appell chains built from classical orthogonal polynomials; testing it on monic Laguerre combinations would be a direct extension.
The exact-count statement suggests a computational certificate: for fixed coefficients, checking the zeros of one finite polynomial $P$ and one small-degree $q_{n_0}$ decides real-rootedness of the whole sequence.
The paper proves convergence of the non-real zeros to the scaled zeros of $P$ but does not give rates; studying their fluctuations as $n$ grows is a natural next step.
The paper leaves an explicit conjecture on the number of positive and negative zeros when all zeros of $P$ are non-real; the conjecture's parity-dependent counts are numerically checkable and, if true, would complete the sign-count description.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a real contribution, not a desk reject. The generalized Hermite construction in Section 3 is the strongest part: the backward shift ΛH_n = H_{n+1} plus the φ,ψ sequences gives a clean framework where interlacing and Turán inequalities fall out naturally. The standard-normalization results, Theorem 1.2 and Corollary 4.1, are convincing and improve the threshold from [8] so that it depends only on the non-real zeros of P. The Appell classification in Theorem 1.3 is genuinely new if it holds.

Soft spots, roughly in order of severity.

(5.6) is load-bearing and imported from the author's unpublished [7]. It is the only mechanism that converts non-real zeros of P into non-real zeros of q_n, and Corollary 5.1 also depends on it through Lemma 2.4. The generating function (5.3) is present, so a referee can reasonably ask for a proof of (5.6) in an appendix or a citation to a published source. I do not think the asymptotic is false, but as written the central Appell results rest on unverified external support. The stress-test's concern lands; I would call it repairable rather than fatal.

Lemma 2.4 is stated with its proof omitted. It is not entirely trivial because the n* must be uniform in θ. It may well be true, but it needs a proof in the paper.

The proof of Theorem 1.3 has structural gaps. The induction on K − N_nr skips the case K − N_nr = 1, and the 'if' direction for N_nr = 0 is dispatched with the line 'because q_n = q'_{n+1}', which by itself is a non sequitur. The claim is true — eventual real-rootedness from (5.6) plus the monotonicity Z_nr(n+1) ≥ Z_nr(n) gives all n — but the written proof does not say that. Repairable, not fatal.

Corollary 4.1 says 'the cases K − 2m ≥ 2 can be proved similarly' and Corollary 5.5 says 'if θ < 0, the proof is similar.' Those are exactly the places a referee will want expansion.

Citation pattern: heavy reliance on [7] and [8], both self-authored preprints. Self-citation is not automatically a problem, but here the unproved asymptotic is the hinge, so the dependence should be made explicit.

This paper is for people working on zeros of orthogonal polynomial combinations, Appell sequences, and real-rootedness. It deserves a serious referee. My verdict is conditional: the Appell results can become solid after the (5.6) gap is closed and the induction is cleaned up; otherwise a major revision is needed.

Referee Report

4 major / 5 minor

Summary. The paper studies the number of real zeros of finite linear combinations of K+1 consecutive Hermite polynomials in two normalizations: the standard normalization and the Appell normalization H_n/(2^n n!). For both cases it introduces the auxiliary polynomial P(x)=∑γ_j x^{K−j} and shows that its zeros determine the real-rootedness of q_n=Σγ_j H_{n−j}. In the standard case (Theorem 1.2), the paper proves that real-rootedness of P implies real-rootedness of q_n for all n≥K, with interlacing, and that if P has nonreal zeros a threshold depending only on those nonreal zeros suffices. In the Appell case (Theorem 1.3), it gives a complete classification: q_n is real-rooted for all n if and only if P has no nonreal zeros, and when P has N_nr nonreal zeros, after a threshold n0 the polynomial q_n has exactly n−N_nr real zeros and N_nr nonreal zeros. The paper also proves Turán-type inequalities, asymptotic zero behavior via Mehler–Heine-type limits, and a conjecture on the signs of the real zeros when P is nonreal-rooted. The main tools are a backward-shift operator Λ f=2xf−f′, a generalization of Hermite polynomials, Obreshkov's interlacing theorem, and a complex asymptotic for the Appell-normalized combinations.

Significance. If the central claims hold, this is a substantial contribution to the zero-location theory of linear combinations of classical orthogonal polynomials. The identification of P as the governing object is natural and cleanly separates the two normalizations: in the standard normalization real-rootedness of P is sufficient, while in the Appell normalization it is also necessary. The interlacing results and the Turán-type inequalities are of independent interest, and the asymptotics in Corollary 5.3 give a fairly complete picture of the zero distribution. The strategy via the backward-shift operator and generalized Hermite polynomials is elegant and yields new structural results, including the connection to multiple Hermite polynomials in Remark 1. The paper is not circular and introduces no fitted parameters; P is an input polynomial. The main reservations are that two load-bearing inputs are imported from the author's own preprints ([7] and [8]) and that a key lemma and some induction cases are stated without proof; these limit the paper's self-containedness but appear repairable.

major comments (4)

[§5, Eq. (5.6)] The asymptotic (5.6), lim_n (z/(n+1))^n n! q_n((n+1)/z) = z^K e^{-z^2/4} P(1/z), uniformly on compact sets, is the sole mechanism in the proof of Theorem 1.3 that turns nonreal zeros of P into nonreal zeros of q_n. It is imported from the author's preprint [7, Theorem 1.1] without a proof or a statement of its hypotheses. The same asymptotic is used in Corollary 5.1 and Corollary 5.3. Since this is load-bearing, please include a proof or a precise, verifiable statement of the required uniformity; without this the main classification is not established within the manuscript.
[Lemma 2.4] Lemma 2.4 is stated with its proof omitted ('the proof is similar to the usual proof for the Hurwitz's Theorem ... and it is omitted'). The uniformity of the integer n* with respect to the real parameter θ is essential for Corollary 5.1, and the lemma is not a standard textbook statement. Please supply a proof or a reference that actually contains this uniformity assertion; a Rouché-type argument may work, but it must be written out.
[Proof of Theorem 1.3] The induction on K−N_nr in the proof of Theorem 1.3 begins with the base case N_nr=K and then states 'Assume next that K − N_nr > 1.' This excludes the case K−N_nr=1, i.e., P having exactly one real zero. That case is needed for the induction itself and for part (1) when K=1. The closing sentence 'If N_nr=0, then all the zeros of q_n has to be real for n≥0, because q_n=q′_{n+1}' does not by itself prove real-rootedness for all n. The induction step should be formulated for K−N_nr≥1 and should spell out how the case K−N_nr=1 reduces to the base case.
[Proof of Corollary 4.1] In the proof of Corollary 4.1, after treating K−2m=0 and K−2m=1, the statement 'The cases K − 2m ≥ 2 can be proved similarly' delegates the iterative interlacing argument to the reader. Since this is the advertised improvement over Corollary 1.1, the iterative use of Lemmas 2.2 and 2.3 should be written out explicitly rather than left to a 'similarly'.

minor comments (5)

[Introduction] The phrase 'tipe II multiple Hermite polynomials' contains a typo; it should be 'type II'.
[§3, Eq. (3.7)] In the definition of φ{l}, the phrase 'que sequence obtained by removing' should be 'the sequence obtained by removing'.
[§5, Eq. (5.1)] After equation (5.1), the symbol H_n is used for the normalized polynomial, which is also the standard notation for the unnormalized Hermite polynomial used in Sections 3 and 4. This overloads notation; consider writing ~H_n for the normalized polynomial throughout.
[Corollary 5.3] The symbol N_− is used for the number of negative real zeros of P, but this is not defined before its first use in the statement of Corollary 5.3. Please define it explicitly.
[Corollary 5.3, proof] In the proof of Corollary 5.3, 'asimptotic' should be 'asymptotic'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: P is an input, not a fitted quantity; the Appell-normalization result relies on the cited asymptotic (5.6), but that is an independent analytic lemma, not an assumption of the conclusion.

full rationale

The derivation is not circular. The polynomial P is defined directly from the coefficients gamma_j, the same coefficients that define q_n, but neither object is defined in terms of the theorem's conclusion; the paper proves structural relations between their zeros. In the standard normalization, the real-rootedness of q_n when P is real-rooted is established self-containedly through the generalized Hermite polynomials h^phi_n (Section 3 and Corollary 4.1), with phi_i the real zeros of P. The eventual real-rootedness in the non-real case reduces to Corollary 1.1 from the author's prior preprint [8], applied to the non-real factor P_nr; that cited result is parameter-free and does not assume the present theorem, so the citation is independent support rather than circularity. In the Appell normalization, the key input is the asymptotic (5.6), quoted from [7, Theorem 1.1]. Although this is a load-bearing self-citation, it is a concrete limit formula that follows from the exact generating function (5.3) and is not equivalent to the real-rootedness classification being proved; it is used through the Hurwitz-type Lemma 2.4 to transfer non-real zeros of P to non-real zeros of q_n. The omitted proof of Lemma 2.4 is a completeness issue, not a circular step. There are no fitted parameters, no prediction renamed as a fit, and no uniqueness or ansatz imported from the authors' prior work to force the choice. The asymptotic and threshold results from [7] and [8] should be independently verified for full reproducibility, but their reliance does not make the argument circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

This is a pure mathematics paper, so the ledger has no fitted free parameters and no invented physical entities. The arbitrary coefficients gamma_j are inputs, not fitted constants, and the threshold n0 is derived rather than chosen by hand. The real load-bearing assumptions are external results imported from the literature, including two self-authored preprints ([7] and [8]) whose proofs are not reproduced here.

assumptions (7)

ad hoc to paper Corollary 1.1 of [8]: finite consecutive Hermite combinations with standard normalization are real-rooted for n at least an explicit bound.
Imported from the author's companion preprint arXiv:2505.11956; it supplies the existence and size of n0 in Corollary 4.1. Not proved in this manuscript.
ad hoc to paper Brenke asymptotic (5.6) from [7, Theorem 1.1]: scaled q_n converges to z^K e^{-z^2/4} P(1/z).
Used to locate non-real zeros of q_n in the Appell normalization; it is the load-bearing input for Theorem 1.3 and Corollaries 5.1 to 5.3.
standard math Obreshkov theorem (Theorem 2.1): interlacing of p and q is equivalent to real-rootedness of all real linear combinations of p and q.
Used repeatedly to convert interlacing statements into real-rootedness statements.
standard math Hurwitz-type stability lemma (Lemma 2.4): non-real zeros of the limit are preserved in fn - theta gn for large n.
The proof is omitted; the statement is a standard Hurwitz argument and is used in Corollary 5.1.
standard math Mehler-Heine formulas (5.19) and (5.20), and the Hermite zero counting limit (5.23).
Used in Corollary 5.3 for central zero asymptotics and for the arcsine limiting distribution of the real zeros.
standard math Beardon-Driver Theorem 5.4: a polynomial in the span of p_r through p_n has zeros in at least r intervals between consecutive zeros of p_n.
Used in Corollary 5.3 to control which zeros of q_n lie near the zeros of Hermite polynomials.
domain assumption Parameter restrictions: gamma_0=1, gamma_K nonzero, psi_i greater than -2, and H_j=0 for j<0 in the Appell section.
These restrictions ensure fixed degree, nonzero leading coefficients, and well-defined behavior for negative-index Hermite polynomials.

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Cite this review

Pith. "Pith review of Zeros of linear combinations of Hermite polynomials." pith.science (2026). https://pith.science/paper/AF3PIE7U

@misc{pith2026250515330,
  author       = {Pith},
  title        = {Pith review of: Zeros of linear combinations of Hermite polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AF3PIE7U}},
  note         = {Machine review of arXiv:2505.15330}
}

abstract

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Hermite polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde H_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\dots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$. We consider two different normalizations of Hermite polynomials: the standard one (i.e. $\tilde H_n=H_n$), and $\tilde H_n=H_n/(2^nn!)$ (so that $q_n$ are Appell polynomials: $q_n'=q_{n-1}$). In both cases, we show the key role played by the polynomial $P(x)=\sum_{j=0}^K\gamma_jx^{K-j}$ to solve this problem. In particular, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Zeros of linear combinations of Laguerre polynomials
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Finite sums of consecutive Laguerre polynomials are real-rooted for large n exactly when an auxiliary coefficient polynomial has only real roots, with four Laguerre normalizations giving different counts of non-real zeros.
Zeros of GKP sequences of polynomials
math.GM 2026-07 accept novelty 6.0 of 10

Under |φ_n|+ψ_n<0, GKP polynomials have real simple interlacing zeros between the roots of the driving quadratic, with explicit extreme-zero asymptotics when ψ is constant.

Reference graph

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Reviewed August 7, 2026 · model on record in the stance chip above.

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P. Br¨ and´ en, On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture, J. Algebraic Combin. 20 (2004), 119–130

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