Positivity of Tur\'an determinants for orthogonal polynomials II

Ryszard Szwarc

REVIEW 2 major objections 4 minor 21 references

Positivity of Tur\'an determinants for orthogonal polynomials II

T0 review · 2 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read A coefficient rule guarantees Turán's inequality on [-1,1].

desk verdict Solid extension of the author's Turán criterion with an honest correction of a prior error; the proof has a minor black-box invocation and a strict-monotonicity wording gap, but the math is sound. read the letter →

arxiv 2009.09711 v3 pith:JUR7JQ6I submitted 2020-09-21 math.CA

Ryszard Szwarc This is my paper · ORCID

classification math.CA MSC 42C0547B39

keywords orthogonalpolynomialsTurándeterminantsinequalityrecurrencerelationnormalizedassociatedPollaczekcoefficientmonotonicity

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

This paper proves a general criterion under which orthogonal polynomials on $[-1,1]$, normalized so that $P_n(1)=1$, satisfy Turán's inequality: $P_n(x)^2 - P_{n-1}(x)P_{n+1}(x) \ge 0$ for every $n \ge 1$ and every $x$ in $[-1,1]$. The criterion is stated directly in terms of the three-term recurrence coefficients $\alpha_n$ and $\gamma_n$: if $\alpha_n$ is strictly increasing and at most $1/2$, $\gamma_n$ is positive and strictly decreasing, $\alpha_n + \gamma_n \le 1$, and two explicit auxiliary inequalities hold, then the inequality follows. The advance over earlier work is that $\gamma_0$ may be less than $1$, which covers associated polynomials and a parameter range of the Pollaczek family that previously fell outside the known results. The proof works with the ratios $g_n = p_{n+1}(1)/p_n(1)$ and reduces the problem to a known theorem by renormalizing the recurrence. A series of corollaries turns the criterion into simple checks for coefficient sequences of the form $1/2 \pm c\,\delta_n$.

What carries the argument

The load-bearing device is the ratio sequence $g_n = p_{n+1}(1)/p_n(1)$. Lemma 1 bounds $g_n$ between $1$ and the difference quotient $(\alpha_{n+1}\gamma_n - \alpha_n\gamma_{n+1})/(\gamma_n - \gamma_{n+1})$, using inequality (6) to propagate the upper bound and inequality (7) to start it. Lemma 2 proves $g_n$ is nonincreasing by composing the maps $f_k(x) = (1/\gamma_k)(1 - \alpha_k/x)$, which are monotone and ordered on the interval where the bounds live. The ratios convert the original coefficients into normalized recurrence coefficients $\tilde{\alpha}_n = \alpha_n/g_{n-1}$ and $\tilde{\gamma}_n = \gamma_n g_n$ satisfying $\tilde{\alpha}_n + \tilde{\gamma}_n = 1$, with $\tilde{\alpha}_n$ nondecreasing and at most $1/2$; the proof then invokes a previously established theorem to conclude Turán's inequality.

What would settle it

For a concrete test, take Example 3 with $a = 1$, so $\alpha_n = 1/2 - 1/(2(n+1))$ and $\gamma_n = 1/2 + 1/(2(n+2))$, build the normalized polynomials $P_n$ from the recurrence with $P_0 = 1$ and $P_1(1) = 1$, and scan $\Delta_n(x) = P_n(x)^2 - P_{n-1}(x)P_{n+1}(x)$ for $n \le 1000$ on a fine grid of $x$ in $[-1,1]$; a single negative value would refute Theorem 1, since this sequence is claimed to satisfy all hypotheses.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is Theorem 1: for a probability measure on $[-1,1]$ with orthogonal polynomials $p_n$ satisfying $x p_n = \gamma_n p_{n+1} + \alpha_n p_{n-1}$, the normalized polynomials $P_n = p_n/p_n(1)$ have nonnegative Turán determinants on the whole interval provided (a) $\alpha_n$ strictly increases and never exceeds $1/2$, (b) $\gamma_n$ is positive and strictly decreasing, (c) $\alpha_n + \gamma_n \le 1$, and the two auxiliary inequalities $\frac{\alpha_n - \alpha_{n-1}}{\alpha_n \gamma_{n-1} - \alpha_{n-1} \gamma_n} \le \frac{\alpha_{n+1} \gamma_n - \alpha_n \gamma_{n+1}}{\gamma_n - \gamma_{n+1}}$ for $n \ge 1$, and $\gamma_0 - \gamma_1 \le \alpha_1 \gamma_0^2$, hold. The conclusion is exactly $P_n(x)^2 - P_{n-1}(x)P_{n+1}(x) \ge 0$ for $-1 \le x \le 1$ and all $n \ge 1$. The paper further shows that the criterion specializes to simple sufficient conditions when the coefficients are perturbations of $1/2$ by a monotone sequence $\delta_n$, and it applies these to symmetric Pollaczek polynomials, including parameter ranges that were incorrectly excluded in the author's earlier paper.

Load-bearing premise

The load-bearing premise is a previously published theorem, quoted but not proved in this paper, that normalized orthogonal polynomials with nondecreasing coefficient at most $1/2$ already satisfy Turán's inequality; if that theorem is false for any of the rescaled coefficient sequences constructed here, the new criterion collapses.

Editorial extensions

If this is right

Any orthogonal polynomial family whose recurrence coefficients satisfy the four conditions, checked on the coefficients alone, immediately satisfies Turán's inequality on the whole orthogonality interval, without computing $p_n(1)$.
Associated polynomials of order $k$, which typically have $\gamma_0 < 1$, now fall inside the criterion; previously only the case $\gamma_0 \ge 1$ was covered.
The symmetric Pollaczek polynomials with $\lambda > a$ satisfy the criterion, correcting a false statement in the earlier paper; with $\lambda \le a$ the old corollary already applied.
Coefficient sequences of the form $\alpha_n = 1/2 - \alpha\delta_n$, $\gamma_n = 1/2 + \gamma\delta_n$ with $\alpha \ge \gamma > 0$ and $\delta_n \downarrow 0$ automatically satisfy the criterion (Corollary 1), giving a wide and easily testable class.
As a side effect, the same normalization produces nonnegative linearization of the polynomials, a property used in harmonic analysis and Banach-algebra approaches to orthogonal polynomials.

Reading between the lines

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

A natural conjecture is that inequality (6) is not merely sufficient but is the sharp 'ratio-monotonicity' condition: the upper bound on $g_n$ in Lemma 1 is precisely what keeps the renormalized $\tilde{\alpha}_n$ below $1/2$, and families where (6) is an equality may be extremal for Turán determinants.
The proof suggests a transfer principle for other positivity questions: any theorem that holds for normalized coefficients with $\tilde{\alpha}_n$ monotone and at most $1/2$ can be pulled back to raw coefficients whenever the ratio sequence $g_n$ is nonincreasing and lies between 1 and the difference quotient; this could be tested on related inequalities such as higher-order Turán determinants.
Since Corollary 1 is stated with $\delta_n$ merely nonincreasing to 0 while the theorem needs strict decrease, one can attempt a limiting argument: approximate a non-strict sequence by strictly decreasing sequences and check whether Turán positivity is closed under such limits; if it is, the corollary's overbreadth is harmless.
The examples with $\delta_n$ of order $1/n$ suggest an asymptotic connection: the bound (7), $\gamma_0 - \gamma_1 \le \alpha_1\gamma_0^2$, is a one-step condition that might be equivalent to the initial ratio $g_0$ lying in the interval where the comparison $f_1 \le f_0$ holds; this could simplify checking the criterion in practice.

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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

Punchline: the paper does exactly what it says—extends the author's earlier Turán criterion to the γ0 < 1 case and corrects a wrong claim about Pollaczek polynomials. The main proof is clean, and the examples are worked in enough detail to be checked. The only real caveats are a black-box citation at the last step and a strict-monotonicity wording issue in the corollaries.

What's new: Theorem 1 gives a sufficient condition on recurrence coefficients α_n, γ_n for Turán's inequality to hold on [-1,1]. Unlike the earlier result in [18], it does not require γ0 ≥ 1. The proof reduces the problem to the normalized polynomials by studying g_n = p_{n+1}(1)/p_n(1). Lemmas 1 and 2 establish the bounds and monotonicity of g_n; the algebra in both checks out. The transformed coefficients \tilde α_n are nondecreasing and bounded above by 1/2, so the conclusion is imported from Theorem 1(i) of [18]. Corollaries 1 and 2 translate the complicated condition (6) into simple monotone parametrizations, and the paper correctly points out that the earlier application to Pollaczek polynomials for λ > a was wrong because γ0 < 1. Examples 2–4 show the extended condition in action.

Soft spots: (1) The reliance on [18] is legitimate—it's a published theorem with an independent proof—but the paper never states the exact hypotheses of Theorem 1(i). If that theorem requires strict increase of α_n, the reduction only gives nondecreasing \tilde α_n; the author should either quote the theorem precisely or note that the conclusion holds under the weaker assumption. (2) Corollaries 1 and 2 write δ_n ց 0, which standardly means nonincreasing, and Theorem 1 calls for strict monotonicity. All the examples use strictly decreasing δ_n, so it's a wording gap rather than a mathematical error, but it should be tightened. (3) The endpoint x = -1 is not discussed explicitly; it follows from the open-interval result by continuity, so that's not a real issue.

Bottom line: this is a solid, honest extension. The self-citation is appropriate—[18] is the natural baseline and the cited result is not being re-proved or hidden. The paper deserves a serious referee; it would pass with minor revisions. I'd send it to review and would cite it in my own work.

Reading group: maybe. It's a narrow result, but the correction to Pollaczek polynomials is worth knowing.

Referee Report

2 major / 4 minor

Summary. The paper proves a sufficient condition, Theorem 1, for Turán's inequality P_n(x)^2 - P_{n-1}(x)P_{n+1}(x) >= 0 on [-1,1] for polynomials orthogonal on [-1,1] and normalized by P_n(1)=1. The condition is expressed directly in terms of the recurrence coefficients α_n and γ_n. The proof introduces g_n = p_{n+1}(1)/p_n(1), proves two lemmas giving bounds and monotonicity of g_n, and then normalizes the recurrence to reduce to Theorem 1(i) of the author's earlier paper [18]. Corollaries 1 and 2 give simpler sufficient conditions of the form α_n = 1/2 - αδ_n, γ_n = 1/2 + γδ_n with δ_n decreasing to 0, and Examples 1-4 apply the results to Pollaczek polynomials and to constructed families.

Significance. If the criterion is correct, it is a useful extension of the author's earlier result to the case γ_0 < 1, which is important for associated polynomials and for Pollaczek polynomials with λ > a. The proof is concise and elementary, with Lemmas 1 and 2 fully proved; the normalization step is transparent and the examples are concrete. The main caveat is that the final step relies on the exact hypotheses and conclusion of Theorem 1(i) of [18], which are not reproduced, and that Corollaries 1 and 2 are stated with nonincreasing δ_n although Theorem 1 requires strict monotonicity. These issues are fixable but affect the statements as written.

major comments (2)

[Theorem 1, proof, final paragraph] The proof of Theorem 1 is not self-contained in its final step: it invokes Theorem 1(i) of [18] without stating that theorem's exact hypotheses or conclusion. If Theorem 1(i) of [18] establishes Turán's inequality only on the open interval (-1,1), then the endpoint x = -1 claimed in Theorem 1 is not covered. If that theorem requires the recurrence coefficients to be strictly increasing, the proof currently only asserts that the normalized coefficients ~α_n are nondecreasing; the strictness does follow from assumption (a) and Lemma 2, but it should be stated explicitly. Please quote the relevant theorem from [18] and verify each of its hypotheses, including the endpoint behavior, before applying it.
[Corollaries 1 and 2] The notation δ_n ց 0 is standardly read as nonincreasing convergence to 0, but Theorem 1(a),(b) require α_n to be strictly increasing and γ_n to be strictly decreasing, which forces δ_n to be strictly decreasing. As written, Corollaries 1 and 2 are overbroad and their proofs do not apply to sequences δ_n with flat parts. Please require strict decrease of δ_n, or add a limiting argument showing that the conclusion persists for nonincreasing δ_n. The examples all use strictly decreasing sequences, so the concrete applications are unaffected.

minor comments (4)

[Corollaries 1 and 2] There are typos: 'satify' should be 'satisfy' in both corollaries, and 'aformentioned' should be 'aforementioned' in Remark 4.
[Corollary 2, proof] The phrase 'Then the conclusion of Corollary 1 holds' is slightly confusing; it should say 'Then the conclusion of Theorem 1 holds' or explicitly state the Turán inequality for the normalized polynomials.
[Example 1 and Remark 2] The references to 'Corollary 1(i) of [18]' and 'Corollary 1(ii) [18]' should be given with complete statements or at least unambiguous labels, since the numbering is easy to confuse with the new Corollary 1 in this paper.
[Throughout] The notation δ_n ց 0 is used in several places; a one-sentence definition of this notation at first use would improve clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the proof reduces to the author's earlier Theorem 1(i) of [18], a published independent theorem, and no fitted parameter or definitional identity is involved.

full rationale

The derivation is not circular in the sense of reducing a claim to its own input by construction. The paper proves Lemmas 1 and 2 directly from the recurrence (5) and hypotheses (6)-(7), obtaining 1 ≤ g_n ≤ (α_{n+1}γ_n − α_nγ_{n+1})/(γ_n − γ_{n+1}) and that g_n is nonincreasing. It then defines the normalized coefficients ~α_n = α_n/g_{n−1} and ~γ_n = γ_n p_{n+1}(1)/p_n(1), checks P_n(1)=1, ~α_n nondecreasing and ≤1/2, and invokes 'Theorem 1(i) of [18]' as a black box. That cited theorem is a published, parameter-free result whose hypotheses (normalized polynomials with increasing α_n ≤ 1/2) do not include the present conditions (6)-(7) or the γ_0 < 1 case; the present contribution is the construction of the normalization and verification of those hypotheses, not a restatement of the conclusion. Thus the central claim does not reduce to its inputs by definition. The main caveats are non-circular: the exact hypotheses of Theorem 1(i) of [18] are not reproduced, so endpoint x = −1 coverage depends on that theorem's precise formulation, and Corollaries 1–2 state δ_n ց 0 without explicitly requiring strict monotonicity, although Theorem 1(a)-(b) assumes strict monotonicity and the examples use strictly decreasing sequences. These are self-containedness and correctness concerns, not circularity; hence score 2 rather than 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new mathematical entities. The proof relies on standard orthogonal polynomial theory and on Theorem 1(i) of [18], which is cited rather than reproved. The parameters α, γ and δ_n in the corollaries are variables in sufficient conditions, not numbers fitted to data.

assumptions (3)

standard math Orthogonal polynomials on [-1,1] satisfy the three-term recurrence (1) with α_0 = 0, p_{-1} = 0 and positive coefficients α_n, γ_n.
Section 1, equation (1). Standard orthogonal polynomial theory via Favard's theorem.
domain assumption For normalized polynomials with recurrence coefficients α_n increasing and α_n ≤ 1/2, Turán's inequality holds (Theorem 1(i) of [18]).
Used at the end of the proof of Theorem 1 as a black box; not proved in this note.
domain assumption The measure µ is symmetric with support [-1,1], so p_n(1) > 0 and normalization P_n = p_n/p_n(1) is well-defined.
Section 1; follows from the roots lying in (-1,1).

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Pith. "Pith review of Positivity of Tur\'an determinants for orthogonal polynomials II." pith.science (2026). https://pith.science/paper/JUR7JQ6I

@misc{pith2026200909711,
  author       = {Pith},
  title        = {Pith review of: Positivity of Tur\'an determinants for orthogonal polynomials II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JUR7JQ6I}},
  note         = {Machine review of arXiv:2009.09711}
}

abstract

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Tur\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Tur\'an's inequality. This extends essentially the results of \cite{szw}. In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.

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Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [18]

Szwarc, Positivity of Tur´ an determinants for orthogonal polynomials

R. Szwarc, Positivity of Tur´ an determinants for orthogonal polynomials. In: Harmonic Analysis and Hypergroups, Eds. K.A. Ross et al., Bi rkhauser, Boston-Basel-Berlin, 1998, 165–182

work page 1998

[2] [1]

Al-Salam, L

W. Al-Salam, L. Carlitz, General Tur´ an expressions forcertain hypergeomet- ric series, Portugal. Math. 16, (1957), 119–127

work page 1957

[3] [2]

Askey, Linearization of the product of orthogonal pol ynomials, Problems in Analysis, R

R. Askey, Linearization of the product of orthogonal pol ynomials, Problems in Analysis, R. Gunning, ed., Princeton University Press, P rinceton, N.J., (1970) 223–228

work page 1970

[4] [3]

Bustoz, M

J. Bustoz, M. E. H. Ismail, Tur´ an inequalities for ultraspherical and continu- ous q–ultraspherical polynomials, SIAM J. Math. Anal. 14 (1983), 807–818

work page 1983

[5] [4]

Bustoz, M

J. Bustoz, M. E. H. Ismail, Tur´ an inequalities for symmetric orthogonal poly- nomials, Internat. J. Math. Math. Sci. 20 (1997), no. 1, 1–7

work page 1997

[6] [5]

Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applications, Vol

T. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applications, Vol. 13, Gordon 6 Breach, New York, 1978

work page 1978

[7] [6]

Gasper, On the extension of Tur´ an’s inequality to Jac obi polynomials, Duke

G. Gasper, On the extension of Tur´ an’s inequality to Jac obi polynomials, Duke. Math. J. 38 (1971), 415–428

work page 1971

[8] [7]

Gasper, An inequality of Tur´ an type for Jacobi polynomials, Proc

G. Gasper, An inequality of Tur´ an type for Jacobi polynomials, Proc. Amer. Math. Soc. 32 (1972), 435–439

work page 1972