Strong Asymptotics for Multiple Laguerre Polynomials

We consider multiple Laguerre polynomials ln 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 ln(nz) are obtained in the entire complex plane by using the Deift-Zhou steepest descent method for a (3 × 3)-matrix Riemann-Hilbert problem.

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