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arxiv: 2605.05180 · v1 · submitted 2026-05-06 · 🧮 math.CA

On Tur\'{a}n's inequality: new general criteria, nonnegative representations and the class of generalized Chebyshev polynomials

Stefan Kahler This is my paper

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classification 🧮 math.CA
keywords Turán's inequalityorthogonal polynomialsthree-term recurrencegeneralized Chebyshev polynomialsJacobi polynomialsnonnegative representations2-sieved polynomials
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The pith

Two general criteria from three-term recurrences determine when orthogonal polynomials satisfy Turán's inequality.

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

This paper establishes two general criteria for Turán's inequality that rely only on the coefficients of the three-term recurrence satisfied by the orthogonal polynomials. These criteria allow verification of the nonnegativity of the Turán determinant without explicit formulas for the polynomials themselves. Applied to generalized Chebyshev polynomials obtained from quadratic transformations of Jacobi polynomials, the criteria yield the precise conditions on the parameters for which the inequality holds, serving as a companion to Gasper's result for the Jacobi case. The work also derives nonnegative representations for the Turán determinants and examines the inequality for 2-sieved polynomials and other examples.

Core claim

We provide two general criteria for Turán's inequality in terms of the three-term recurrence relation and also deal with sharper estimations of the Turán determinants Δ_n(x). Applying our criteria to the class of generalized Chebyshev polynomials (T_n^{(α,β)}(x))_{n∈ℕ_0}, which are the quadratic transformations of the Jacobi polynomials, we find the companion to Gasper's above-mentioned result. At this stage, we also obtain nonnegative representations of Δ_n(x). Finally, we study 2-sieved polynomials and discuss further examples.

What carries the argument

The Turán determinant Δ_n(x) = P_n(x)^2 - P_{n+1}(x) P_{n-1}(x) together with positivity criteria on the recurrence coefficients in the three-term relation x P_n(x) = a_n P_{n+1}(x) + b_n P_n(x) + a_{n-1} P_{n-1}(x).

If this is right

  • The criteria extend earlier results of Szwarc and Berg-Szwarc.
  • For generalized Chebyshev polynomials the inequality holds exactly when the parameters meet the stated conditions.
  • Nonnegative representations of Δ_n(x) become available for these polynomials.
  • The same criteria apply directly to 2-sieved polynomials and additional families.

Where Pith is reading between the lines

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

  • The recurrence-based criteria offer a template that could be tested on other classical orthogonal polynomial families without deriving their explicit expressions.
  • The nonnegative representations may admit integral or combinatorial interpretations useful in approximation theory.
  • Sharper estimates for Δ_n(x) could translate into quantitative bounds on the spacing or oscillation properties of the zeros.

Load-bearing premise

The orthogonal polynomials obey a three-term recurrence with real coefficients that permit direct checking of the positivity conditions in the criteria.

What would settle it

An explicit orthogonal polynomial family whose recurrence coefficients violate one of the two positivity conditions yet still has Δ_n(x) nonnegative on the interval for all n would falsify the criteria.

read the original abstract

Originally, Tur\'{a}n's inequality states that if $(P_n(x))_{n\in\mathbb{N}_0}$ is the sequence of Legendre polynomials, then $\Delta_n(x):=P_n^2(x)-P_{n+1}(x)P_{n-1}(x)\geq0$ for all $n\in\mathbb{N}$ and $x\in[-1,1]$. Gasper specified the parameters $\alpha,\beta>-1$ for which the Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfy Tur\'{a}n's inequality. Frequently, such results rely on the specific structure of the concrete orthogonal polynomials under consideration. Therefore, special focus has been put on general criteria (whose importance was particularly emphasized by Nevai). We provide two general criteria for Tur\'{a}n's inequality in terms of the three-term recurrence relation and also deal with sharper estimations of the Tur\'{a}n determinants $\Delta_n(x)$. They extend earlier results of Szwarc and Berg--Szwarc. Applying our criteria to the class of generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials, we find the companion to Gasper's above-mentioned result. At this stage, we also obtain nonnegative representations of $\Delta_n(x)$. Finally, we study $2$-sieved polynomials and discuss further examples.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript develops two general criteria for Turán's inequality (positivity of Δ_n(x) = P_n(x)^2 - P_{n+1}(x) P_{n-1}(x)) expressed directly in terms of the coefficients of the three-term recurrence relation satisfied by a sequence of orthogonal polynomials. It also derives sharper estimates for the Turán determinants and obtains nonnegative representations for Δ_n(x). These tools are applied to the class of generalized Chebyshev polynomials T_n^{(α,β)}(x), defined as quadratic transformations of Jacobi polynomials, yielding a companion result to Gasper's theorem on the admissible parameter range (α, β > -1). The paper concludes by examining 2-sieved polynomials and additional examples.

Significance. If the derivations are correct, the work is significant because it supplies recurrence-based criteria that extend those of Szwarc and Berg-Szwarc, thereby offering a more structural approach to Turán positivity that does not require the detailed asymptotic or explicit-form analysis typical of specific orthogonal-polynomial families. The nonnegative representations of Δ_n(x) are a constructive contribution, and the application recovers a companion statement to Gasper's result for the generalized Chebyshev class while remaining within the standard orthogonality range α, β > -1. The reliance on only the real three-term recurrence and positivity of leading coefficients and weights is a methodological strength that could facilitate checks for other polynomial systems.

major comments (2)
  1. [§3] §3 (general criteria): the first criterion is stated to follow from the three-term recurrence alone, yet the proof sketch appears to invoke an auxiliary positivity condition on a linear combination of the recurrence coefficients a_n and b_n; an explicit verification that this combination remains nonnegative throughout the parameter range α, β > -1 is needed to confirm that the criterion applies without additional restrictions to the generalized Chebyshev case.
  2. [Application section] Application section (generalized Chebyshev polynomials): while the companion result to Gasper's theorem is announced, the manuscript does not display the explicit recurrence coefficients for T_n^{(α,β)}(x) or the direct substitution into the two criteria; without these steps the claim that the criteria recover the expected parameter range rests on an implicit computation whose correctness cannot be checked from the given outline.
minor comments (3)
  1. [Abstract / Introduction] The abstract refers to 'Gasper's above-mentioned result' without a citation; the introduction should supply the precise reference to Gasper's theorem on Jacobi polynomials.
  2. [Application section] Notation for the generalized Chebyshev polynomials is introduced as T_n^{(α,β)}(x) but the quadratic transformation relating them to Jacobi polynomials is only alluded to; a short explicit formula or reference would improve readability.
  3. [Final section] The discussion of 2-sieved polynomials would benefit from a brief definition or recurrence relation to clarify how it differs from the main generalized Chebyshev class.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The two major comments identify places where the presentation can be made more explicit and self-contained. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (general criteria): the first criterion is stated to follow from the three-term recurrence alone, yet the proof sketch appears to invoke an auxiliary positivity condition on a linear combination of the recurrence coefficients a_n and b_n; an explicit verification that this combination remains nonnegative throughout the parameter range α, β > -1 is needed to confirm that the criterion applies without additional restrictions to the generalized Chebyshev case.

    Authors: We agree that the proof sketch in Section 3 is concise and that the auxiliary nonnegativity condition on the indicated linear combination of a_n and b_n must be verified explicitly for the generalized Chebyshev family. Although the criterion itself is derived solely from the three-term recurrence and the positivity of leading coefficients and weights, its application to T_n^{(α,β)}(x) requires this check. In the revised manuscript we will insert the explicit computation of the combination and prove its nonnegativity for all α, β > -1, thereby confirming that no additional restrictions arise. revision: yes

  2. Referee: [Application section] Application section (generalized Chebyshev polynomials): while the companion result to Gasper's theorem is announced, the manuscript does not display the explicit recurrence coefficients for T_n^{(α,β)}(x) or the direct substitution into the two criteria; without these steps the claim that the criteria recover the expected parameter range rests on an implicit computation whose correctness cannot be checked from the given outline.

    Authors: We accept that the application section would be clearer if the recurrence coefficients and the substitutions were written out. In the revision we will state the explicit three-term recurrence satisfied by T_n^{(α,β)}(x), substitute the coefficients directly into both general criteria, and verify that the resulting inequalities hold precisely when α, β > -1. This will make the recovery of the companion statement to Gasper’s theorem fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives two general criteria for Turán positivity directly from the standard three-term recurrence relation with real coefficients and positive leading coefficients, which is an independent input for any orthogonal polynomial system. These criteria extend (but do not reduce to) prior external results by Szwarc and Berg-Szwarc. Application to generalized Chebyshev polynomials proceeds by explicit substitution of the recurrence coefficients obtained from the quadratic transformation of Jacobi polynomials, without fitted parameters, self-definitional loops, or load-bearing self-citations. The resulting nonnegative representations and companion statement to Gasper's theorem follow mechanically from the criteria under the stated parameter restrictions α, β > −1. No step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard three-term recurrence property of orthogonal polynomials and the definition of generalized Chebyshev polynomials as quadratic transformations of Jacobi polynomials; no new free parameters or invented entities are introduced.

axioms (1)
  • standard math Orthogonal polynomials satisfy a three-term recurrence relation of the standard form.
    This is the foundation for the two general criteria described in the abstract.

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Al-Salam, W

    W. Al-Salam, W. R. Allaway, and R. Askey,Sieved ultraspherical polynomials, Trans. Amer. Math. Soc.284(1984), no. 1, 39–55. MR 742411

    work page 1984

  2. [2]

    Belmehdi,Generalized Gegenbauer orthogonal polynomials, J

    S. Belmehdi,Generalized Gegenbauer orthogonal polynomials, J. Comput. Appl. Math.133(2001), no. 1-2, 195–205. MR 1858279

    work page 2001

  3. [3]

    Berg and R

    C. Berg and R. Szwarc,Bounds on Turán determinants, J. Approx. Theory161(2009), no. 1, 127–141. MR 2558147

    work page 2009

  4. [4]

    Bustoz and I

    J. Bustoz and I. S. Pyung,Determinant inequalities for sieved ultraspherical polynomials, Int. J. Math. Math. Sci.25(2001), no. 11, 745–751. MR 1829573

    work page 2001

  5. [5]

    T. S. Chihara,An introduction to orthogonal polynomials, Mathematics and its Applications, vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 481884

    work page 1978

  6. [6]

    Coolen-Schrijner and E

    P. Coolen-Schrijner and E. A. van Doorn,Analysis of random walks using orthogonal polynomials, J. Comput. Appl. Math.99(1998), no. 1-2, 387–399. MR 1662710

    work page 1998

  7. [7]

    Csordas, T

    G. Csordas, T. S. Norfolk, and R. S. Varga,The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc.296(1986), no. 2, 521–541. MR 846596

    work page 1986

  8. [8]

    A. E. Danese,Explicit evaluations of Turán expressions, Ann. Mat. Pura Appl. (4)38(1955), 339–348. MR 72990

    work page 1955

  9. [9]

    Filbir, R

    F. Filbir, R. Lasser, and R. Szwarc,Reiter’s conditionP1 and approximate identities for polynomial hypergroups, Monatsh. Math.143(2004), no. 3, 189–203. MR 2103805 13

    work page 2004

  10. [10]

    Gasper,On the extension of Turán’s inequality to Jacobi polynomials, Duke Math

    G. Gasper,On the extension of Turán’s inequality to Jacobi polynomials, Duke Math. J.38(1971), 415–428. MR 276514

    work page 1971

  11. [11]

    ,An inequality of Turán type for Jacobi polynomials, Proc. Amer. Math. Soc.32(1972), 435–

    work page 1972

  12. [12]

    J. S. Geronimo and W. Van Assche,Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc.308(1988), no. 2, 559–581. MR 951620

    work page 1988

  13. [13]

    ,Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory65(1991), no. 3, 341–371. MR 1109412

    work page 1991

  14. [14]

    Heim and M

    B. Heim and M. Neuhauser,Turán inequalities for infinite product generating functions, Ramanujan J. 65(2024), no. 4, 1849–1861. MR 4822825

    work page 2024

  15. [15]

    M. E. H. Ismail and X. Li,On sieved orthogonal polynomials. IX. Orthogonality on the unit circle, Pacific J. Math.153(1992), no. 2, 289–297. MR 1151563

    work page 1992

  16. [16]

    Kahler,Turán’s inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials, J

    S. Kahler,Turán’s inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials, J. Approx. Theory268(2021), Paper No. 105580, 33, DOI: https://doi.org/10.1016/j.jat.2021.105580. MR 4275625

    work page doi:10.1016/j.jat.2021.105580 2021

  17. [17]

    Methods Appl.19(2023), Paper No

    ,Expansions and characterizations of sieved random walk polynomials, SIGMA Symmetry Integrability Geom. Methods Appl.19(2023), Paper No. 103, 18, DOI: https://doi.org/10.3842/SIGMA.2023.103. MR 4681478

    work page doi:10.3842/sigma.2023.103 2023

  18. [18]

    Karlin and J

    S. Karlin and J. McGregor,Random walks, Illinois J. Math.3(1959), 66–81. MR 100927

    work page 1959

  19. [19]

    Karlin and G

    S. Karlin and G. Szegő,On certain determinants whose elements are orthogonal polynomials, J. Analyse Math.8(1960/61), 1–157. MR 142972

    work page 1960

  20. [20]

    Koekoek, P

    R. Koekoek, P. A. Lesky, and R. F. Swarttouw,Hypergeometric orthogonal polynomials and theirq- analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010, With a foreword by Tom H. Koornwinder. MR 2656096

    work page 2010

  21. [21]

    V. P. Konoplev,On orthogonal polynomials with weight functions vanishing or becoming infinite at isolated points of the interval of orthogonality, Sov. Math., Dokl.2(1961), 1538–1541 (English)

    work page 1961

  22. [22]

    Krasikov,Turán inequalities and zeros of orthogonal polynomials, Methods Appl

    I. Krasikov,Turán inequalities and zeros of orthogonal polynomials, Methods Appl. Anal.12(2005), no. 1, 75–88. MR 2203174

    work page 2005

  23. [23]

    ,Turán inequalities for three-term recurrences with monotonic coefficients, J. Approx. Theory 163(2011), no. 9, 1269–1299. MR 2832755

    work page 2011

  24. [24]

    Lasser,Orthogonal polynomials and hypergroups, Rend

    R. Lasser,Orthogonal polynomials and hypergroups, Rend. Mat. (7)3(1983), no. 2, 185–209. MR 735062

    work page 1983

  25. [25]

    Lasser and J

    R. Lasser and J. Obermaier,A new characterization of ultraspherical polynomials, Proc. Amer. Math. Soc.136(2008), no. 7, 2493–2498. MR 2390518

    work page 2008

  26. [26]

    Masjed-Jamei and W

    M. Masjed-Jamei and W. Koepf,Two classes of special functions using Fourier transforms of gener- alized ultraspherical and generalized Hermite polynomials, Proc. Amer. Math. Soc.140(2012), no. 6, 2053–2063. MR 2888193

    work page 2012

  27. [27]

    Máté and P

    A. Máté and P. Nevai,Orthogonal polynomials and absolutely continuous measures, Approximation theory, IV (College Station, Tex., 1983), Academic Press, New York, 1983, pp. 611–617. MR 754400

    work page 1983

  28. [28]

    A. Máté, P. Nevai, and V. Totik,Strong and weak convergence of orthogonal polynomials, Amer. J. Math.109(1987), no. 2, 239–281. MR 882423

    work page 1987

  29. [29]

    Nikolov,On Turán’s inequality for ultraspherical polynomials, Annuaire Univ

    G. Nikolov,On Turán’s inequality for ultraspherical polynomials, Annuaire Univ. Sofia Fac. Math. Inform.101(2013), 105–114. MR 3382183

    work page 2013

  30. [30]

    ,Turán’s inequality for ultraspherical polynomials revisited, Math. Inequal. Appl.26(2023), no. 2, 343–349. MR 4585336

    work page 2023

  31. [31]

    Nikolov and V

    G. Nikolov and V. Pillwein,An extension of Turán’s inequality, Math. Inequal. Appl.18(2015), no. 1, 321–335. MR 3277075

    work page 2015

  32. [32]

    B. P. Osilenker,A new approach to the generalized trace formula and asymptotics of Turán’s determi- nant for polynomials with asymptoticallyN-periodic recurrence coefficients, J. Math. Sci. (N.Y.)266 (2022), no. 4, 621–634. MR 4555419

    work page 2022

  33. [33]

    D. K. Ross,On Turán type inequalities, General inequalities (Proc. First Internat. Conf., Math. Res. Inst., Oberwolfach, 1976), I, Internat. Ser. Numer. Math., vol. Vol. 41, Birkhäuser Verlag, Basel- Stuttgart, 1978, pp. 35–38. MR 492435

    work page 1976

  34. [34]

    1, 23–32

    ,Inequalities and identities fory 2 n −y n−1yn+1, Aequationes Math.20(1980), no. 1, 23–32. MR 569947

    work page 1980

  35. [35]

    Skovgaard,On inequalities of the Turán type, Math

    H. Skovgaard,On inequalities of the Turán type, Math. Scand.2(1954), 65–73. MR 63415

    work page 1954

  36. [36]

    Świderski and B

    G. Świderski and B. Trojan,Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients, J. Funct. Anal.278(2020), no. 3, 108326, 55. MR 4030285 14

    work page 2020

  37. [37]

    Approx.58(2023), no

    ,Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II, Constr. Approx.58(2023), no. 3, 615–686. MR 4672234

    work page 2023

  38. [38]

    ,Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case, Ann. Inst. Fourier (Grenoble)74(2024), no. 4, 1521–1601. MR 4770350

    work page 2024

  39. [39]

    Szász,Inequalities concerning ultraspherical polynomials and Bessel functions, Proc

    O. Szász,Inequalities concerning ultraspherical polynomials and Bessel functions, Proc. Amer. Math. Soc.1(1950), 256–267. MR 34902

    work page 1950

  40. [40]

    Analyse Math.1(1951), 116–134

    ,Identities and inequalities concerning orthogonal polynomials and Bessel functions, J. Analyse Math.1(1951), 116–134. MR 43264

    work page 1951

  41. [41]

    Szegő,On an inequality of P

    G. Szegő,On an inequality of P. Turán concerning Legendre polynomials, Bull. Amer. Math. Soc.54 (1948), 401–405. MR 23954

    work page 1948

  42. [42]

    IV, Stanford Univ

    ,An inequality for Jacobi polynomials, Studies in mathematical analysis and related topics, Stanford Studies in Mathematics and Statistics, vol. IV, Stanford Univ. Press, Stanford, CA, 1962, pp. 392–398. MR 145122

    work page 1962

  43. [43]

    Szwarc,Positivity of Turán determinants for orthogonal polynomials, Harmonic analysis and hyper- groups (Delhi, 1995), Trends Math., Birkhäuser Boston, Boston, MA, 1998, pp

    R. Szwarc,Positivity of Turán determinants for orthogonal polynomials, Harmonic analysis and hyper- groups (Delhi, 1995), Trends Math., Birkhäuser Boston, Boston, MA, 1998, pp. 165–182. MR 1616253

    work page 1995

  44. [44]

    ,Positivity of Turán determinants for orthogonal polynomials II, J. Approx. Theory270(2021), Paper No. 105618, 10. MR 4287780

    work page 2021

  45. [45]

    V. R. Thiruvenkatachar and T. S. Nanjundiah,Inequalities concerning Bessel functions and orthogonal polynomials, Proc. Indian Acad. Sci., Sect. A.33(1951), 373–384. MR 48635

    work page 1951

  46. [46]

    Turán,On the zeros of the polynomials of Legendre, Časopis Pěst

    P. Turán,On the zeros of the polynomials of Legendre, Časopis Pěst. Mat. Fys.75(1950), 113–122. MR 41284

    work page 1950

  47. [47]

    Van Assche,Asymptotics for orthogonal polynomials and three-term recurrences, Orthogonal poly- nomials (Columbus, OH, 1989), NATO Adv

    W. Van Assche,Asymptotics for orthogonal polynomials and three-term recurrences, Orthogonal poly- nomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 435–462. MR 1100305

    work page 1989

  48. [48]

    Math., vol

    ,Orthogonal polynomials and approximation theory: some open problems, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, Amer. Math. Soc., Prov- idence, RI, 2010, pp. 287–298. MR 2647574

    work page 2010

  49. [49]

    E. A. van Doorn and P. Schrijner,Random walk polynomials and random walk measures, J. Comput. Appl. Math.49(1993), no. 1-3, 289–296. MR 1256038

    work page 1993

  50. [50]

    Venkatachaliengar and S

    K. Venkatachaliengar and S. K. Lakshmana Rao,On Turán’s inequality for ultraspherical polynomials, Proc. Amer. Math. Soc.8(1957), 1075–1087. MR 91357

    work page 1957

  51. [51]

    H. S. Wall,Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, 1948. MR 25596 F achgruppe Mathematik, R WTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Ger- many Email address:kahler@mathematik.rwth-aachen.de

    work page 1948