Strong asymptotics for Jacobi-Piñeiro orthogonal polynomials

Abstract

We investigate the asymptotic behavior of Jacobi-Piñeiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x)=x^{{\alpha }_j}(1-x{)}^{\beta }$, $j=1,2$ where ${\alpha }_1,{\alpha }_2,\beta >-1$, and ${\alpha }_1-{\alpha }_2\notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3\times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

56 pages