Riemann-Hilbert problem associated with Angelesco systems
Let \(\Delta_1=[-\lambda,-1]\) and \(\Delta_2=[1,\lambda]\) with a real number \(\lambda>1\). For \(j=1,2\), let \(h_j\) be a holomorphic function in a neighbourhood \(\mathcal V_j\) of \(\Delta_j\) that does not vanish on \(\mathcal V_j\) and is positive on \(\Delta_j\). Consider the Angelesco system \((\sigma_1,\sigma_2)\) of measures defined by \[ d\sigma_j(x)=\frac{h_j(x)dx}{\sqrt{(\lambda-|x|)(|x|-1)}},\quad x\in\Delta_j,\;j=1,2. \] The main goal of this paper is to study the strong asymptotics of the sequence of multiple orthogonal polynomials \(\{Q_{\mathbf{n}}:\;\text{\textbf{n}}=(n,n),\;n\in\mathbb Z_{+}\}\) with deg\,\(Q_{\mathbf{n}}\leq 2n\), determined by the conditions \[ \int_{\Delta_j}x^{\nu}Q_{\mathbf{n}}(x)d\sigma_j(x)=0,\quad \nu=0,\dots,n,\;j=1,2. \] The authors' methods rely upon analysing the Riemann-Hilbert problem for multiple orthogonal polynomials, which was introduced in [\textit{W. Van Assche}, \textit{J. S. Geronimo} and \textit{A. B. J. Kuijlaars}, NATO Sci. Ser. II, Math. Phys. Chem. 30, 23--59 (2001; Zbl 0997.42012)].
scientific article; zbMATH DE number 5624871