Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 033, 41 pages      arXiv:2411.04206      https://doi.org/10.3842/SIGMA.2025.033

Uniformity of Strong Asymptotics in Angelesco Systems

Maxim L. Yattselev
Department of Mathematical Sciences, Indiana University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202, USA

Received November 08, 2024, in final form April 28, 2025; Published online May 08, 2025

Abstract
Let $ \mu_1 $ and $ \mu_2 $ be two complex-valued Borel measures on the real line such that $ \operatorname{supp} \mu_1 =[\alpha_1,\beta_1]$ < $\operatorname{supp} \mu_2 =[\alpha_2,\beta_2] $ and $ {\rm d}\mu_i(x) = -\rho_i(x){\rm d}x/2\pi {\rm i}$, where $ \rho_i(x) $ is the restriction to $ [\alpha_i,\beta_i] $ of a function non-vanishing and holomorphic in some neighborhood of $ [\alpha_i,\beta_i] $. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $ (n_1,n_2) $ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $ \min\{n_1,n_2\} $.

Key words: multiple orthogonal polynomials; Angelesco systems; strong asymptotics; Riemann-Hilbert analysis.

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