Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane
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Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane Gernot Akemann1 · Taro Nagao2 · Iván Parra1 · Graziano Vernizzi3 Received: 24 May 2019 / Revised: 8 April 2020 / Accepted: 7 May 2020 © The Author(s) 2020
Abstract We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polyno(1+α) (z) for α > −1 containing the Legendre polynomials Pn (z) and the subset mials Cn (α+ 1 ,± 1 )
Pn 2 2 (z) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit α → ∞ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices. Keywords Planar orthogonal polynomials · Ellipse · Bergman space · Selberg integral
Communicated by Peter Forrester.
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Gernot Akemann [email protected] Taro Nagao [email protected] Iván Parra [email protected] Graziano Vernizzi [email protected]
1
Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany
2
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
3
Siena College, 515 Loudon Road, Loudonville, NY 12211, USA
123
Constructive Approximation
Mathematics Subject Classification 30C10 · 33C45 · 42C05 · 15B52
1 Introduction Orthogonal polynomials in the complex plane play an important role for non-Hermitian random matrix theory. A prominent example is the elliptic Ginibre ensemble with complex normal matrix elements, having different variances for their real and imaginary parts, [23]. Its complex eigenvalues follow a determinantal point process, with its kernel constituted by the Hermite polynomials orthogonal in the complex plane [9]. Likewise, the chiral partner of this ensemble leads to a kernel of generalised Laguerre polynomials orthogonal in the complex plane [2,21] and [13]. The respective kernels allow for a complete characterisation of all complex eigenvalue correlation functions of these ensembles of random matrices. Moreover, in the limit of weak non-Hermiticity introduced in [8], these nontrivial polynomials allow us to study an interpolation between the statistics of real eigenvalues of Hermitian random matrices on the one hand, e.g., of the Gaussian Unitary Ensemble characterised by Hermite polynomials on the real line, and those of complex eigenvalues, e.g., of the Ginibre ensemble, being characterised by monomial polynomials in the complex plane. We refer to [3] for a list of interpolating limiting kern
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