Semiclassical Approximation with Compex Phases for Constructing Effective Plancherel-Rotach type asymptotics of 1-D and 2-D orthogonal polynomials

Many orthogonal polynomials $u(n,z)$, ($n$ is the number of the polynomial, $z$ is its argument), for example, the Chebyshev, Hermite, Laguerre, Legendre, are determined by recurrent relations (or difference equations) of the second order. For large numbers of $n$, they are approximated by the exponent, trigonometric, or special functions of a complex argument. For example, Hermite polynomials are approximated by the Plancherel-Rotach formulas, in which the special function is the Airy function Ai, Legendre polynomials are approximated by the zero-order Bessel function, and so on. We discuss an approach [1] to finding asymptotics of this type that are uniform in the variable $z$, based on the transition from discrete equations to continuous pseudodifferential equations in the variable $x=nh$, for functions $w(x,z)$, ($u(k,z)=w(kh,z)$, where $h \sim O (1/n)$ is an artificial small parameter) and the subsequent application of the semiclassical approximation with complex phases to them. The developed approach is generalized for 2-D Hermitian type orthogonal polynomials $H(n_1,n_2,z,a)$ with two indices $n_1,n_2$. This part of the talk contains the results recently obtained together with A. I. Aptekarev and D. N. Tulyakov [2,3].
The work was supported by Government program №AAAAA20-120011690131-7.
The talk is based on joint work with Prof. Anna Tsvetkova.
Bibliography
[1] A. I. Aptekarev, S. Yu. Dobrokhotov, D. N. Tulyakov, A. V. Tsvetkova, Plancherel-Rotach type asymptotics for multiple orthogonal Hermite polynomials and recurrence relations, Izvestiya: Mathematics 86:1, 32-91.
[2] S. Yu. Dobrokhotov and A. V. Tsvetkova, An Approach to Finding the Asymptotics of Polynomials Given by Recurrence Relations, Russian Journal of Mathematical Physics, Vol. 28, No. 2, 2021, pp. 198-223.
[3] S. Yu. Dobrokhotov and A. V. Tsvetkova, Asymptotics of multiple orthogonal Hermite polynomials $H(n_1,n_2,z,\alpha)$ determined by a third-order differential equation, Russian Journal of Mathematical Physics, Vol. 28, No. 4, 2021, pp. 439-454.