Approximation for the zeros of generalized Hermite polynomials via modulated elliptic function

V. Yu. Novokshenov

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 27, Tome 494 (2020), pp. 228-241

Résumé

Distributiond of zeros of polynomials constitute a classic analytic problem. In the paper, a distribution of zeroes to generalized Hermite polynomials $H_{m,n}(z)$ is approximated as $m$, $n\to\infty$, $m/n=O(1)$. These polyno-\break mials defined as Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices. The calcualation is based on scaling reduction of Painlevé IV equation which has solutions $u(z)= -2z +\partial_z \ln H_{m,n+1}(z)/H_{m+1,n}(z)$. For large $m, n$ the logarithmic derivative of $H_{m,n}$ satisfies equation for elliptic Weierstrass function with slowly varying coefficients. In this scaling limit the zeros coincide with poles of such modulated Weierstrass function, and a stability in linear limit gives estimates for the set od zeros.This construction is relatively simple and avoids bulky calculations by isomonodromic deformation method.