Asymptotics of Multiple Orthogonal Hermite Polynomials \(H_{n_1,n_2}(z,\alpha)\) Determined by a Third-Order Differential Equation

Abstract

In the paper, we study the asymptotics of multiple orthogonal Hermite polynomials \(H_{n_1,n_2}(z,\alpha)\) that are determined by orthogonality relations with respect to two weights that are Gaussian exponents with shifted maxima. These polynomials can be defined using recurrence relations, and also, as shown by A. I. Aptekarev, A. Branquinho, and W. Van Assche, as certain solutions to a third-order differential equation. Starting from this differential equation, we obtain asymptotics of such polynomials as \(|n|=\sqrt{n_1^2+n_2^2} \rightarrow \infty\) in the form of the Airy function \({\rm Ai}\) and its derivative \({\rm Ai}'\) of a compound argument.