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 polynomials C_n^{(1+alpha )}(z) for alpha >-1 containing the Legendre polynomials P_n(z) and the subset P_n^{(alpha +frac{1}{2},pm frac{1}{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 alpha rightarrow infty 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.
Highlights
Orthogonal polynomials in the complex plane play an important role for non-Hermitian random matrix theory
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 show that the classical Gegenbauer or ultraspherical polynomials Cn(1+α)(z), for α > −1, provide a family of planar orthogonal polynomials on the interior of an ellipse parametrised by h(z) := (Re z)2/a2 + (Im z)2/b2, with a > b > 0 and weight function (1 − h(z))α
Summary
Orthogonal polynomials in the complex plane play an important role for non-Hermitian random matrix theory. We show that the classical Gegenbauer or ultraspherical polynomials Cn(1+α)(z), for α > −1, provide a family of planar orthogonal polynomials on the interior of an ellipse parametrised by h(z) := (Re z)2/a2 + (Im z)2/b2, with a > b > 0 and weight function (1 − h(z))α They generalise the monomials that appear in the determinantal point process on the unit disc, obtained from the ensemble of truncated unitary random matrices [27]. In “Appendix C” an alternative orthogonality proof for Gegenbauer polynomials independent of the degree is given which in contrast relies on the known orthogonality of the Chebyshev polynomials of the second kind on the unweighted ellipse. We present the Selberg integral based on the family of Gegenbauer polynomials Cn(1+α)(z) as an example that can be analytically continued in α
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