Real-valued semiclassical approximation for the asymptotics with complex-valued phases and its application to multiple orthogonal Hermite polynomials

The multiple orthogonal Hermite polynomials $H_{n_1,n_2}(z,a)$ are defined by the following recurrence relations (see [1]):
\begin{gather*} H_{n_1+1,n_2}(z,a)=(z+a)H_{n_1,n_2}(z,a)-\frac{1}{2} \left(n_1H_{n_1-1,n_2}(z,a)+n_2H_{n_1,n_2-1}(z,a)\right),\\ H_{n_1,n_2+1}(z,a)=(z-a)H_{n_1,n_2}(z,a)-\frac{1}{2} \left(n_1H_{n_1-1,n_2}(z,a)+n_2H_{n_1,n_2-1}(z,a)\right). \end{gather*}

We construct the uniform Plancherel–Rotach-type asymptotics of diagonal polynomials $H_{n,n}(z,a)$ as $n \rightarrow \infty$. To obtain the result we develop the method which we call “real-valued semiclassical approximation for the asymptotics with complex-valued phases” (another approach based on the construction of decompositions of bases of difference equations was recently developed by A. I. Aptekarev and D. N. Tulyakov).

The discussed approach can be applied to construct asymptotics for a wide class of orthogonal polynomials. The idea of the method is to introduce a small artificial parameter $h \sim \frac{1}{n}$ and a continuous function $\varphi(x,z,a)$ such that $H_{n,n}(z,a)=\varphi(nh,z,a)$. This allows us to reduce the system that defines the polynomials to a pseudo-differential equation for $\varphi$, where $x$ is a variable and $(z,a)$ are parameters. Seeking its solution in the WKB-form, one obtains the Hamiltonian-Jacobi equation with the complex-valued Hamiltonian connected with the third-order algebraic curve. In general case, to obtain the result for such problems, a transition from real variable $x$ to a complex one is made. In this problem, we propose a different approach based on a reduction of the original problem to three equations, two of which have asymptotics with purely imaginary phases, and the symbol of the third one has the form $\cos p + V_0(x) +h V_1(x)+O(h^2).$ The construction of an asymptotic solution of the last equation is described in [2], what allows us to obtain a uniform asymptotics for $H_{n,n}(z,a)$ in the form of the Airy function ${\rm Ai}$ of the real-valued argument.

The talk is based on the joint work with A. I. Aptekarev (Keldysh Institute of Applied Mathematics RAS), S. Yu. Dobrokhotov (Ishlinsky Institute for Problems in Mechanics RAS) and D. N. Tulyakov (Keldysh Institute of Applied Mathematics RAS).