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

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.