## Universality at an endpoint for orthogonal polynomials with Geronimus-type weights

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## Abstract:

We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with respect to a Geronimus-type weight on an arc of the unit circle. The key tool is a formula of McLaughlin for the $n^{th}$ power of a $2\times 2$ matrix, which we use to derive convenient formulas for Geronimus polynomials.## References

- Peter Borwein and Tamás Erdélyi,
*Polynomials and polynomial inequalities*, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR**1367960**, DOI 10.1007/978-1-4612-0793-1 - P. Bourgade,
*On random matrices and $L$-functions*, Ph.D. Thesis, New York University, available at http://www.cims.nyu.edu/$\sim$bourgade/papers/PhDThesis.pdf. - María José Cantero and Arieh Iserles,
*From orthogonal polynomials on the unit circle to functional equations via generating functions*, Trans. Amer. Math. Soc.**368**(2016), no. 6, 4027–4063. MR**3453364**, DOI 10.1090/tran/6454 - M. S. Costa, R. L. Lamblém, J. H. McCabe, and A. Sri Ranga,
*Para-orthogonal polynomials from constant Verblunsky coefficients*, J. Math. Anal. Appl.**426**(2015), no. 2, 1040–1060. MR**3314878**, DOI 10.1016/j.jmaa.2015.02.005 - Tivadar Danka,
*Universality limits for generalized Jacobi measures*, Adv. Math.**316**(2017), 613–666. MR**3672915**, DOI 10.1016/j.aim.2017.06.026 - T. Danka and V. Totik,
*Christoffel functions with power type weights*, to appear in Journal of the European Mathematical Society. - Jeroen Demeyer,
*Recursively enumerable sets of polynomials over a finite field*, J. Algebra**310**(2007), no. 2, 801–828. MR**2308181**, DOI 10.1016/j.jalgebra.2006.09.030 - Leonid Golinskii,
*Akhiezer’s orthogonal polynomials and Bernstein-Szegő method for a circular arc*, J. Approx. Theory**95**(1998), no. 2, 229–263. MR**1652884**, DOI 10.1006/jath.1998.3197 - Leonid Golinskii, Paul Nevai, Ferenc Pintér, and Walter Van Assche,
*Perturbation of orthogonal polynomials on an arc of the unit circle. II*, J. Approx. Theory**96**(1999), no. 1, 1–32. MR**1659428**, DOI 10.1006/jath.1998.3208 - Leonid Golinskii, Paul Nevai, and Walter Van Assche,
*Perturbation of orthogonal polynomials on an arc of the unit circle*, J. Approx. Theory**83**(1995), no. 3, 392–422. MR**1361537**, DOI 10.1006/jath.1995.1128 - M. Bello Hernández and E. Miña Díaz,
*Strong asymptotic behavior and weak convergence of polynomials orthogonal on an arc of the unit circle*, J. Approx. Theory**111**(2001), no. 2, 233–255. MR**1849548**, DOI 10.1006/jath.2001.3574 - 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 - William B. Jones, Olav Njåstad, and W. J. Thron,
*Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle*, Bull. London Math. Soc.**21**(1989), no. 2, 113–152. MR**976057**, DOI 10.1112/blms/21.2.113 - B. G. Korenev,
*Bessel functions and their applications*, Analytical Methods and Special Functions, vol. 8, Taylor & Francis Group, London, 2002. Translated from the Russian by E. V. Pankratiev. MR**1963816** - A. B. J. Kuijlaars and M. Vanlessen,
*Universality for eigenvalue correlations from the modified Jacobi unitary ensemble*, Int. Math. Res. Not.**30**(2002), 1575–1600. MR**1912278**, DOI 10.1155/S1073792802203116 - D. S. Lubinsky,
*Universality limits at the hard edge of the spectrum for measures with compact support*, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 099, 39. MR**2439541**, DOI 10.1093/imrp/rnn099 - D. S. Lubinsky,
*A new approach to universality limits at the edge of the spectrum*, Integrable systems and random matrices, Contemp. Math., vol. 458, Amer. Math. Soc., Providence, RI, 2008, pp. 281–290. MR**2411912**, DOI 10.1090/conm/458/08941 - Doron S. Lubinsky and Vy Nguyen,
*Universality limits involving orthogonal polynomials on an arc of the unit circle*, Comput. Methods Funct. Theory**13**(2013), no. 1, 91–106. MR**3089946**, DOI 10.1007/s40315-013-0011-5 - Attila Máté, Paul Nevai, and Vilmos Totik,
*Szegő’s extremum problem on the unit circle*, Ann. of Math. (2)**134**(1991), no. 2, 433–453. MR**1127481**, DOI 10.2307/2944352 - J. Mc Laughlin,
*Combinatorial identities deriving from the $n$th power of a $2\times 2$ matrix*, Integers**4**(2004), A19, 15. MR**2116004** - Lynsey Naugle,
*Orthogonal Polynomials on an Arc of the Unit Circle with Respect to a Generalized Jacobi Weight: A Riemann-Hilbert Method Approach*, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–The University of Mississippi. MR**3698864** - Ferenc Janos Pinter,
*Perturbation of orthogonal polynomials on an arc of the unit circle*, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–The Ohio State University. MR**2692359** - F. Pintér and P. Nevai,
*Schur functions and orthogonal polynomials on the unit circle*, Approximation theory and function series (Budapest, 1995) Bolyai Soc. Math. Stud., vol. 5, János Bolyai Math. Soc., Budapest, 1996, pp. 293–306. MR**1432676** - Brian Simanek,
*Two universality results for polynomial reproducing kernels*, J. Approx. Theory**216**(2017), 16–37. MR**3612482**, DOI 10.1016/j.jat.2017.01.002 - Barry Simon,
*Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line*, J. Approx. Theory**126**(2004), no. 2, 198–217. MR**2045539**, DOI 10.1016/j.jat.2003.12.002 - Barry Simon,
*Orthogonal polynomials on the unit circle. Part 2*, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR**2105089**, DOI 10.1090/coll/054.2/01 - Barry Simon,
*Orthogonal polynomials on the unit circle. Part 2*, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR**2105089**, DOI 10.1090/coll/054.2/01 - Barry Simon,
*The Christoffel-Darboux kernel*, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 295–335. MR**2500498**, DOI 10.1090/pspum/079/2500498 - Herbert Stahl and Vilmos Totik,
*General orthogonal polynomials*, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR**1163828**, DOI 10.1017/CBO9780511759420

## Additional Information

**Brian Simanek**- Affiliation: Department of Mathematics, Baylor University, One Bear Place 97328, Waco, Texas 76798
- MR Author ID: 959574
- Received by editor(s): August 18, 2017
- Received by editor(s) in revised form: December 18, 2017
- Published electronically: April 26, 2018
- Communicated by: Mourad Ismail
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 3995-4007 - MSC (2010): Primary 42C05; Secondary 33C45
- DOI: https://doi.org/10.1090/proc/14085
- MathSciNet review: 3825852