Approximation of Zeros of Generalized Hermite Polynomials by Modulated Elliptic Function

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