# Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane

Akemann G, Nagao T, Parra I, Vernizzi G (2019)
arXiv:1905.02397.

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Preprint | Englisch
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Abstract / Bemerkung
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 $P_n^{(\alpha+\frac12,\pm\frac12)}(z)$ 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 first up to fourth kind. The limit $\alpha\to\infty$ 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.
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arXiv:1905.02397
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Akemann G, Nagao T, Parra I, Vernizzi G. Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane. arXiv:1905.02397. 2019.
Akemann, G., Nagao, T., Parra, I., & Vernizzi, G. (2019). Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane. arXiv:1905.02397
Akemann, G., Nagao, T., Parra, I., and Vernizzi, G. (2019). Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane. arXiv:1905.02397.
Akemann, G., et al., 2019. Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane. arXiv:1905.02397.
G. Akemann, et al., “Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane”, arXiv:1905.02397, 2019.
Akemann, G., Nagao, T., Parra, I., Vernizzi, G.: Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane. arXiv:1905.02397. (2019).
Akemann, Gernot, Nagao, Taro, Parra, Iván, and Vernizzi, G. “Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane”. arXiv:1905.02397 (2019).

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arXiv: 1905.02397