Painlevé VI and Hankel determinants for the generalized Jacobi weight

2009/08/31 by Dan Dai, D Dai, L Zhang +1
Mathematics · Physics and Astronomy · #Algebra over a field #Derivative (finance) #Hankel matrix #Jacobi polynomials #Mathematical functions and polynomials #Monic polynomial #Nonlinear Waves and Solitons #Orthogonal polynomials #Orthogonality #Quantum Mechanics and Non-Hermitian Physics #math.CA

paper · pdf · doi:10.1088/1751-8113/43/5/055207

20 pages. Revised version with some modifications. Typos corrected, reference updated

arxiv created 2009/11/30 · openalex publication_date 2010/01/14 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We study the Hankel determinant of the generalized Jacobi weight (x-t)^γx^α(1-x)^β for x∈[0,1] with α, β>0, t < 0 and γ∈ℝ. Based on the ladder operators for the corresponding monic orthogonal polynomials P_n(x), it is shown that the logarithmic derivative of Hankel determinant is characterized by a τ-function for the Painlev'e VI system.

Citations

• Jacobi polynomials from compatibility conditions
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