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  "slug": "logarithmic-stenger-matrix-stability",
  "title": "Positive stability of logarithmic Stenger integration matrices: a sharp forward boundary, reverse instability and a certified counterexample",
  "shortTitle": "When logarithmic Stenger integration matrices stay stable",
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  "oneLine": "Integration matrices built on Gaussian nodes for the weight xᵅ[log(1/x)]ᵐ are stable at every size exactly when α ≤ 1 when integrating forward; integrating backward fails for m > 1, and an exact certificate at α = −9/10, m = 1, n = 13 disproves the reverse clause of a 2019 Gautschi–Hairer conjecture as worded.",
  "abstract": "Integration matrices replace indefinite integration by multiplication of point values; spectral, Sinc-type and collocation methods built on them need every eigenvalue to have positive real part. Gautschi and Hairer (Journal of Inequalities and Applications, 2019) tested Stenger's conjecture that Gaussian nodes guarantee this and stated five refined conjectures. This paper studies the restricted Stenger matrices Uₙ (integrate from 0) and Vₙ (integrate to 1) whose nodes are the zeros of the orthogonal polynomials for xᵅ[log(1/x)]ᵐ on (0,1), with unweighted integrals. Uₙ is positive stable at every degree exactly when −1 < α ≤ 1; for each fixed α > 1 the number of unstable eigenvalues tends to infinity, and the first threshold θₙ is larger than one at every degree and tends to one. For m > 1, Vₙ is unstable at every large degree with a diverging count; for 0 < m ≤ 1 and α ≥ m/2 − 1 it is stable at every degree. At m = 1, an exact rational Routh certificate shows that V₁₃ has exactly two eigenvalues with negative real part when α = −9/10, disproving the reverse clause of Gautschi–Hairer Conjecture 5.3 as worded; exact counts suggest that α = −1/2 is the true boundary, which remains a conjecture. The mechanism is a pair of locally uniform limits of the scaled characteristic polynomials to the Kummer functions ₁F₁(1; α+1; −z) and ₁F₁(1; m+1; −z), whose zeros cross into the left half-plane exactly when the exponent exceeds one. The same method answers the first-threshold questions of Conjectures 5.1, 5.2 and 5.5 for Jacobi and Laguerre weights: the stability range in 5.2 is sharp; under the table-consistent Jacobi-exponent reading of 5.1 the first thresholds are larger than one, finite for every n ≥ 5 and tend to one; in 5.5 they are larger than one and tend to one, so the uniform reading of 5.5 is false. It also shows that the associated collocation Runge–Kutta methods eventually fail A-stability for α > 1, m ≥ 1, and proves that the symmetric Jacobi threshold at degree five is exactly 9. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-10-22",
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      "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review."
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  "keywords": [
    "Stenger conjecture",
    "spectral integration matrices",
    "positive stability",
    "orthogonal polynomials",
    "logarithmic weights",
    "Kummer functions",
    "Gaussian quadrature",
    "collocation Runge–Kutta methods"
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    "Sharp forward boundary: for every m > 0, Uₙ is positive stable at every degree if and only if −1 < α ≤ 1; for each fixed α > 1 the number of unstable eigenvalues tends to infinity, and the first threshold θₙ(m) is larger than 1 at every degree and tends to 1 (Theorem 2.2, Corollary 2.3, proof).",
    "Reverse matrices: for m > 1, Vₙ is unstable at every large degree with a diverging count; for 0 < m ≤ 1 and α ≥ m/2 − 1 it is stable at every degree (Theorem 2.4(i)–(ii), proof).",
    "Counterexample: at m = 1, α = −9/10, n = 13, Vₙ has exactly two eigenvalues with negative real part and none on the axis, disproving the reverse clause of Gautschi–Hairer Conjecture 5.3 as worded (Theorem 2.4(iii), exact rational certificate). Exact onset degrees 13, 15, 17, 22 for α = −9/10, −4/5, −7/10, −3/5; the exact reverse boundary α = −1/2 is Conjecture 7.1 (open).",
    "Mechanism: the scaled characteristic polynomials of Uₙ and Vₙ, at z/n², converge to ₁F₁(1; α+1; −z) and ₁F₁(1; m+1; −z) locally uniformly; the zeros of ₁F₁(1; γ+1; −z) lie in the open left half-plane for γ > 1 and on the imaginary axis for γ = 1 (Theorem 2.1, Lemma 5.1, proof).",
    "Jacobi and Laguerre weights: all-degree boundary at exponent 1. Conjecture 5.2 holds iff β ≤ 1 (sharp). Under the Jacobi-exponent reading of Conjecture 5.1 the first thresholds exceed 1, are finite for every n ≥ 5 and tend to 1; whether the fixed-degree stable set is an interval is not decided. For Conjecture 5.5 the thresholds exceed 1, are finite for n ≥ 5 and tend to 1, so its uniform reading is false. The symmetric Jacobi threshold at n = 5 is exactly 9 (Theorem 6.1, Corollary 6.2, Lemma 6.3 on sections of the exponential series, Proposition 6.4).",
    "Collocation Runge–Kutta methods with these nodes are not A-stable for large n when α > 1 and m ≥ 1 (Corollary 2.5, proof)."
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    {
      "citation": "Gautschi and Hairer (2019), On conjectures of Stenger in the theory of orthogonal polynomials. Journal of Inequalities and Applications 2019, art. 159 (source of Conjectures 5.1–5.5).",
      "url": "https://doi.org/10.1186/s13660-019-2107-6",
      "doi": "10.1186/s13660-019-2107-6"
    },
    {
      "citation": "Ait-Haddou and Alselami (2023), The Stenger conjectures and the A-stability of collocation Runge–Kutta methods. Journal of Inequalities and Applications 2023, art. 107 (forward stability for non-increasing weights, including −1 < α ≤ 0; the Jacobi stable side).",
      "url": "https://doi.org/10.1186/s13660-023-03019-8",
      "doi": "10.1186/s13660-023-03019-8"
    },
    {
      "citation": "Charalambides (2014), A general stability theorem with applications. Proceedings of the American Mathematical Society 142, 191–197.",
      "url": "https://doi.org/10.1090/S0002-9939-2013-11731-4",
      "doi": "10.1090/S0002-9939-2013-11731-4"
    },
    {
      "citation": "Stenger (2011), Handbook of Sinc Numerical Methods. CRC Press.",
      "url": "https://doi.org/10.1201/b10375",
      "doi": "10.1201/b10375"
    },
    {
      "citation": "Hairer and Wanner (1996), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed. Springer (collocation Runge–Kutta methods).",
      "url": "https://doi.org/10.1007/978-3-642-05221-7",
      "doi": "10.1007/978-3-642-05221-7"
    },
    {
      "citation": "Gautschi (2010), Gauss quadrature routines for two classes of logarithmic weight functions. Numerical Algorithms 55, 265–277.",
      "url": "https://doi.org/10.1007/s11075-010-9366-0",
      "doi": "10.1007/s11075-010-9366-0"
    },
    {
      "citation": "Conway and Deift (2018), Asymptotics of polynomials orthogonal with respect to a logarithmic weight. SIGMA 14, 056.",
      "url": "https://doi.org/10.3842/SIGMA.2018.056",
      "doi": "10.3842/SIGMA.2018.056"
    },
    {
      "citation": "NIST Digital Library of Mathematical Functions, Section 13.9: zeros of Kummer functions.",
      "url": "https://dlmf.nist.gov/13.9",
      "doi": null
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