Strong Asymptotics for Multiple Laguerre Polynomials

We consider multiple Laguerre polynomials l n of degree 2n orthogonal on (0,∞) with respect to the weights $x^{\alpha}e^{-\beta_{1}x}$ and $x^{\alpha}e^{-\beta_{2}x}$ , where -1 < α, 0 < β1 < β2, and we study their behavior in the large n limit. The analysis differs among three different cases which correspond to the ratio β2/β1 being larger, smaller, or equal to some specific critical value κ. In this paper, the first two cases are investigated and strong uniform asymptotics for the scaled polynomials l n (nz) are obtained in the entire complex plane by using the Deift-Zhou steepest descent method for a (3 × 3)-matrix Riemann-Hilbert problem.