Geodesic
Approximation for the zeros of generalized Hermite polynomials via modulated elliptic function
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 27, Tome 494 (2020), pp. 228-241 
V. Yu. Novokshenov. Approximation for the zeros of generalized Hermite polynomials via modulated elliptic function. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 27, Tome 494 (2020), pp. 228-241. http://geodesic.mathdoc.fr/item/ZNSL_2020_494_a10/
@article{ZNSL_2020_494_a10,
     author = {V. Yu. Novokshenov},
     title = {Approximation for the zeros of generalized {Hermite} polynomials via modulated elliptic function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {228--241},
     year = {2020},
     volume = {494},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_494_a10/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

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