Uniformity of Strong Asymptotics in Angelesco Systems

Let ${\mu }_1$ and ${\mu }_2$ be two complex-valued Borel measures on the real line such that $\mathrm{supp}{\mu }_1=[{\alpha }_1,{\beta }_1]<\mathrm{supp}{\mu }_2=[{\alpha }_2,{\beta }_2]$ and $\mathrm{d}{\mu }_i(x)=-{\rho }_i(x)\mathrm{d}x\mathrm{/}2\pi \mathrm{i}$, 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 $\min \{n_1,n_2\}$.