Bouali, M.; Hamouda, M. S.; Al Jeaid, Hind K. Zeros of generalized Hermite polynomials and ensemble of Gaussian random matrices. (English) Zbl 1501.15025 Math. Notes 112, No. 1, 40-58 (2022). The paper investigates the global asymptotic eigenvalue density of fixed-trace generalized Gaussian ensemble of random matrices. The asymptotic behavior of the density of the eigenvalues coincides with the limit density of the zeros of some orthogonal polynomials. The orthogonal polynomials considered in this article are generalized Hermite polynomials.The considered generalized Hermite polynomials \(H_{m}^\lambda (x)\) are given by \[ H_{2m}^\lambda (x) =(-1)^m\frac{(2m)!}{m!}\sum_{k-0}^m (-1)^k \left(\begin{matrix} m\\ k\end{matrix} \right) \frac{\Gamma(\lambda +1/2)}{\Gamma(k+\lambda +1/2)} x^{2k} \] and \[ H_{2m+1}^\lambda (x) =(-1)^m\frac{(2m+1)!}{m!}\sum_{k-0}^m (-1)^k \left(\begin{matrix} m\\ k\end{matrix} \right) \frac{\Gamma(\lambda +1/2)}{\Gamma(k+\lambda +3/2)} x^{2k+1}. \] They are orthogonal with respect to the generalized Gaussian measure \[ \mu(dx)=|x|^{2\lambda}e^{-x^2}dx. \] The authors study the global behavior of the zeros of these polynomials and consider the probability measure \[ \nu_{n,\lambda}=\frac{1}{n}\sum_{i=1}^n \delta_{x_i^{(n,\lambda)}}, \] where \(x_{i}^{(n,\lambda)}, \; i=1,2, \dots, n\), are the zeros of \(H_n^\lambda\).Using scaled versions of this measure, the authors derive results concerning the convergence of probability measures and show that the asymptotic behavior of certain probability measures depends on the asymptotic behavior of the zeros of the generalized Hermite polynomials. Reviewer: Ahmed I. Zayed (Chicago) MSC: 15B52 Random matrices (algebraic aspects) 15B57 Hermitian, skew-Hermitian, and related matrices 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 60B20 Random matrices (probabilistic aspects) Keywords:orthogonal polynomials; probability measures; logarithmic potential PDFBibTeX XMLCite \textit{M. Bouali} et al., Math. Notes 112, No. 1, 40--58 (2022; Zbl 1501.15025) Full Text: DOI References: [1] Forrester, P., Log-Gases and Random Matrices (2010), Princeton University Press · Zbl 1217.82003 [2] Hiai, F.; Petz, D., The Semicircle Law, Free Random Variables and Entropy (2000), American Mathematical Society · Zbl 0955.46037 [3] I. Dumitriu, “Eigenvalue statistics for beta-ensembles,” Ph. D thesis, Department of Mathematics, MIT, No. (2003). 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