Scholarlink

Abstract <p>In this paper, our aim is to investigate orthogonal polynomials and Hankel determinants that are generated by a symmetric nonsingular Jacobi weight. The methods utilized, such as ladder operators and Coulomb fluid, significantly contribute to a more profound comprehension of the properties of the ensemble and their relationships with well-established mathematical frameworks. By adapting the ladder operators to the monic orthogonal polynomials concerning this weight and carefully monitoring the evolution of parameters in the orthogonality relation, we discover a significant connection between one of the auxiliary quantities <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{f}_{n}}(t)\)</EquationSource> <!--ComMat2570165Dan-m1--> </InlineEquation> and the Painlevé V equation, following a suitable transformation of variables. Through the utilization of the Coulomb fluid, we derive the large <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--ComMat2570165Dan-m2--> </InlineEquation> asymptotic expansion of the recurrence coefficient, aiding in the reduction of the second-order differential equation satisfied by the monic orthogonal polynomials associated with this weight to the analogous general Heun equation. Furthermore, we identify a novel quantity linked to the logarithmic derivative of the Hankel determinant that satisfies both a differential equation and a difference equation. These analyses allow us to establish connections among diverse mathematical entities and offer a more profound insight into the underlying mathematical structures at play.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Painlevé, Heun and the Symmetric Nonsingular Jacobi Polynomials

  • 期刊论文

  • 发表日期：

    2025年12月1日

查看全文
  • Dan Wang

摘要

Abstract

In this paper, our aim is to investigate orthogonal polynomials and Hankel determinants that are generated by a symmetric nonsingular Jacobi weight. The methods utilized, such as ladder operators and Coulomb fluid, significantly contribute to a more profound comprehension of the properties of the ensemble and their relationships with well-established mathematical frameworks. By adapting the ladder operators to the monic orthogonal polynomials concerning this weight and carefully monitoring the evolution of parameters in the orthogonality relation, we discover a significant connection between one of the auxiliary quantities \({{f}_{n}}(t)\) and the Painlevé V equation, following a suitable transformation of variables. Through the utilization of the Coulomb fluid, we derive the large \(n\) asymptotic expansion of the recurrence coefficient, aiding in the reduction of the second-order differential equation satisfied by the monic orthogonal polynomials associated with this weight to the analogous general Heun equation. Furthermore, we identify a novel quantity linked to the logarithmic derivative of the Hankel determinant that satisfies both a differential equation and a difference equation. These analyses allow us to establish connections among diverse mathematical entities and offer a more profound insight into the underlying mathematical structures at play.