Szegő recurrence for multiple orthogonal polynomials on the unit circle

Kozhan, Rostyslav; Vaktnäs, Marcus

doi:10.1090/proc/16811

Szegő recurrence for multiple orthogonal polynomials on the unit circle
HTML articles powered by AMS MathViewer

by Rostyslav Kozhan and Marcus Vaktnäs;

Proc. Amer. Math. Soc. 152 (2024), 2983-2997

DOI: https://doi.org/10.1090/proc/16811

Published electronically: May 22, 2024

HTML | PDF | Request permission

Abstract:

We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegő recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel–Darboux formula. These results should be viewed as the direct analogue of the nearest neighbour recurrence relations from the theory of multiple orthogonal polynomials on the real line.

References

C. Álvarez-Fernández, U. Fidalgo Prieto, and M. Mañas, Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy, Adv. Math. 227 (2011), no. 4, 1451–1525., DOI 10.1016/j.aim.2011.03.008
A. I. Aptekarev, Multiple orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 423–447. MR 1662713, DOI 10.1016/S0377-0427(98)00175-7
Alexander I. Aptekarev, Sergey A. Denisov, and Maxim L. Yattselev, Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials, Trans. Amer. Math. Soc. 373 (2020), no. 2, 875–917. MR 4068253, DOI 10.1090/tran/7959
Alexander I. Aptekarev, Sergey A. Denisov, and Maxim L. Yattselev, Jacobi matrices on trees generated by Angelesco systems: asymptotics of coefficients and essential spectrum, J. Spectr. Theory 11 (2021), no. 4, 1511–1597. MR 4349670, DOI 10.4171/jst/380
Alexander I. Aptekarev and Rostyslav Kozhan, Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a Nevai class, J. Approx. Theory 255 (2020), 105409, 21. MR 4083626, DOI 10.1016/j.jat.2020.105409
Alexei Borodin, Biorthogonal ensembles, Nuclear Phys. B 536 (1999), no. 3, 704–732. MR 1663328, DOI 10.1016/S0550-3213(98)00642-7
Ruymán Cruz-Barroso, Carlos Díaz Mendoza, and Ramón Orive, Multiple orthogonal polynomials on the unit circle. Normality and recurrence relations, J. Comput. Appl. Math. 284 (2015), 115–132. MR 3319498, DOI 10.1016/j.cam.2014.11.004
E. Daems and A. B. J. Kuijlaars, A Christoffel-Darboux formula for multiple orthogonal polynomials, J. Approx. Theory 130 (2004), no. 2, 190–202., DOI 10.1016/j.jat.2004.07.003
E. Daems and A. B. J. Kuijlaars, Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions, J. Approx. Theory 146 (2007), no. 1, 91–114., DOI 10.1016/j.jat.2006.12.001
Sergey A. Denisov and Maxim L. Yattselev, Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality, Adv. Math. 396 (2022), Paper No. 108114, 79. MR 4370474, DOI 10.1016/j.aim.2021.108114
M. Duits, On global fluctuations for non-colliding processes, Ann. Probab. 46 (2018), no. 3, 1279–1350., DOI 10.1214/17-AOP1185
Maurice Duits, Benjamin Fahs, and Rostyslav Kozhan, Global fluctuations for multiple orthogonal polynomial ensembles, J. Funct. Anal. 281 (2021), no. 5, Paper No. 109062, 49. MR 4253930, DOI 10.1016/j.jfa.2021.109062
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
A.B.J. Kuijlaars, Multiple orthogonal polynomial ensembles, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., Amer. Math. Soc., Providence, RI 507 (2010), 155–176.
A. B. J. Kuijlaars, A. Martínez-Finkelshtein, and F. Wielonsky, Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights, Comm. Math. Phys. 286 (2009), no. 1, 217–275. MR 2470930, DOI 10.1007/s00220-008-0652-9
Andrei Martínez-Finkelshtein and Walter Van Assche, What is…a multiple orthogonal polynomial?, Notices Amer. Math. Soc. 63 (2016), no. 9, 1029–1031. MR 3525716, DOI 10.1090/noti1430
A. Martínez-Finkelshtein and G. L. F. Silva, Spectral curves, variational problems and the Hermitian matrix model with external source, Comm. Math. Phys. 383 (2021), no. 3, 2163–2242., DOI 10.1007/s00220-021-03999-y
Judit Mínguez Ceniceros and Walter Van Assche, Multiple orthogonal polynomials on the unit circle, Constr. Approx. 28 (2008), no. 2, 173–197. MR 2358391, DOI 10.1007/s00365-007-0680-2
F. Ndayiragije and W. Van Assche, Asymptotics for the ratio and the zeros of multiple Charlier polynomials, J. Approx. Theory 164 (2012), no. 6, 823–840., DOI 10.1016/j.jat.2012.03.005
T. Neuschel and W. Van Assche, Asymptotic zero distribution of Jacobi-Piñeiro and multiple Laguerre polynomials, J. Approx. Theory 205 (2016), 114–132., DOI 10.1016/j.jat.2016.02.002
E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas., DOI 10.1090/mmono/092
Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
Grzegorz Świderski and Walter Van Assche, Christoffel functions for multiple orthogonal polynomials, J. Approx. Theory 283 (2022), Paper No. 105820, 22. MR 4483325, DOI 10.1016/j.jat.2022.105820
Walter Van Assche, Nearest neighbor recurrence relations for multiple orthogonal polynomials, J. Approx. Theory 163 (2011), no. 10, 1427–1448. MR 2832734, DOI 10.1016/j.jat.2011.05.003
W. Van Assche, Ratio asymptotics for multiple orthogonal polynomials, Modern trends in constructive function theory, Contemp. Math., Amer. Math. Soc., Providence, RI 661 (2016), 73–85.