Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

Amílcar Branquinho, Ana Foulquié Moreno, Assil Fradi and Manuel Mañas
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Amílcar Branquinho: Departamento de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal
Ana Foulquié Moreno: Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal
Assil Fradi: Mathematical Physics Special Functions and Applications Laboratory, The Higher School of Sciences and Technology of Hammam Sousse, University of Sousse, Sousse 4002, Tunisia
Manuel Mañas: Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain

Mathematics, 2022, vol. 10, issue 8, 1-25

Abstract: In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.

Keywords: Riemann–Hilbert problems; matrix Pearson equations; matrix biorthogonal polynomials; discrete integrable systems; non-Abelian discrete Painlevé IV equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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