Differential equation for Jacobi-Pineiro polynomials

2005/11/05 by E. Mukhin, Mukhin, E., Alexander Varchenko +2
Mathematics · Physics and Astronomy · #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.QA

paper · pdf · doi:10.48550/arxiv.math/0511138

Latex, 9 pages, revised version, misprints are corrected

openalex publication_date 2005/11/05 · arxiv created 2007/04/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For r∈ \Z_{≥ 0}, we present a linear differential operator %(\di)^r+1+ a₁(x)(\di)^r+...+a_r+1(x) of order r+1 with rational coefficients and depending on parameters. This operator annihilates the r-multiple Jacobi-Piñeiro polynomial. For integer values of parameters satisfying suitable inequalities, it is the unique Fuchsian operator with kernel consisting of polynomials only and having three singular points at x=0, 1, ∞ with arbitrary non-negative integer exponents 0, m₁+1, >..., m₁+...+m_r+r at x=0, special exponents 0, k+1, k+2,..., k+r at x=1 and arbitrary exponents at x=∞.

Citations

• Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture

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