Szegő polynomials from hypergeometric functions

Ranga, A.

doi:10.1090/S0002-9939-2010-10592-0

Szegő polynomials from hypergeometric functions
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by A. Sri Ranga PDF

Proc. Amer. Math. Soc. 138 (2010), 4259-4270 Request permission

Abstract:

Szegő polynomials with respect to the weight function $\omega (\theta ) = e^{\eta \theta } [\sin (\theta /2)]^{2\lambda }$, where $\eta , \lambda \in \mathbb {R}$ and $\lambda > -1/2$ are considered. Many of the basic relations associated with these polynomials are given explicitly. Two sequences of para-orthogonal polynomials with explicit relations are also given.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Gábor Szegő, Collected papers. Vol. 1, Contemporary Mathematicians, Birkhäuser, Boston, Mass., 1982. 1915–1927; Edited by Richard Askey; Including commentaries and reviews by George Pólya, P. C. Rosenbloom, Askey, L. E. Payne, T. Kailath and Barry M. McCoy. MR 674482
A. Cachafeiro, F. Marcellán, and C. Pérez, Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein-Szegö measure, Adv. Comput. Math. 26 (2007), no. 1-3, 81–104. MR 2350346, DOI 10.1007/s10444-004-7644-x
Ruymán Cruz-Barroso, Pablo González-Vera, and F. Perdomo-Pío, Quadrature formulas associated with Rogers-Szegő polynomials, Comput. Math. Appl. 57 (2009), no. 2, 308–323. MR 2488385, DOI 10.1016/j.camwa.2008.10.068
Leyla Daruis, Olav Njåstad, and Walter Van Assche, Szegő quadrature and frequency analysis, Electron. Trans. Numer. Anal. 19 (2005), 48–57. MR 2149269
George Gasper, Orthogonality of certain functions with respect to complex valued weights, Canadian J. Math. 33 (1981), no. 5, 1261–1270. MR 638380, DOI 10.4153/CJM-1981-095-3
Ya.L. Geronimus, “Orthogonal Polynomials”, Amer. Math. Soc. Transl., Ser. 2, vol. 108, American Mathematical Society, Providence, RI, 1977.
Leonid Golinskii and Andrej Zlatoš, Coefficients of orthogonal polynomials on the unit circle and higher-order Szegő theorems, Constr. Approx. 26 (2007), no. 3, 361–382. MR 2335688, DOI 10.1007/s00365-006-0650-7
E. Hendriksen and H. van Rossum, Orthogonal Laurent polynomials, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 1, 17–36. MR 834317, DOI 10.1016/1385-7258(86)90003-X
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
William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
Lisa Lorentzen and Haakon Waadeland, Continued fractions with applications, Studies in Computational Mathematics, vol. 3, North-Holland Publishing Co., Amsterdam, 1992. MR 1172520
A. L. Lukashov and F. Peherstorfer, Zeros of polynomials orthogonal on two arcs of the unit circle, J. Approx. Theory 132 (2005), no. 1, 42–71. MR 2110575, DOI 10.1016/j.jat.2004.10.009
A. Martínez-Finkelshtein, K. T.-R. McLaughlin, and E. B. Saff, Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics, Constr. Approx. 24 (2006), no. 3, 319–363. MR 2253965, DOI 10.1007/s00365-005-0617-6
J. Petronilho, Orthogonal polynomials on the unit circle via a polynomial mapping on the real line, J. Comput. Appl. Math. 216 (2008), no. 1, 98–127. MR 2421843, DOI 10.1016/j.cam.2007.04.024
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
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, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1 (2007), no. 4, 713–772. MR 2350223, DOI 10.3934/ipi.2007.1.713
G. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, Math. Z. 6 (1920), no. 3-4, 167–202 (German). MR 1544404, DOI 10.1007/BF01199955
G. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, Math. Z. 9 (1921), no. 3-4, 167–190 (German). MR 1544462, DOI 10.1007/BF01279027
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
N. M. Temme, Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986), no. 4, 369–376. MR 892162, DOI 10.1007/BF01893438
Satoshi Tsujimoto and Alexei Zhedanov, Elliptic hypergeometric Laurent biorthogonal polynomials with a dense point spectrum on the unit circle, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 033, 30. MR 2506179, DOI 10.3842/SIGMA.2009.033