Bannai-Ito polynomials and dressing chains

Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei

doi:10.1090/S0002-9939-2014-12165-4

Bannai-Ito polynomials and dressing chains
HTML articles powered by AMS MathViewer

by Maxim Derevyagin, Satoshi Tsujimoto, Luc Vinet and Alexei Zhedanov PDF

Proc. Amer. Math. Soc. 142 (2014), 4191-4206 Request permission

Abstract:

Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials – the complementary Bannai-Ito polynomials – are shown to arise in the framework of the SDG maps.

References

F. V. Atkinson, Multiparameter eigenvalue problems, Mathematics in Science and Engineering, Vol. 82, Academic Press, New York-London, 1972. Volume I: Matrices and compact operators. MR 0451001
Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
Włodzimierz Bryc, Wojciech Matysiak, and Jacek Wesołowski, Quadratic harnesses, $q$-commutations, and orthogonal martingale polynomials, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5449–5483. MR 2327037, DOI 10.1090/S0002-9947-07-04194-3
Włodek Bryc and Jacek Wesołowski, Askey-Wilson polynomials, quadratic harnesses and martingales, Ann. Probab. 38 (2010), no. 3, 1221–1262. MR 2674998, DOI 10.1214/09-AOP503
M. I. Bueno and F. Marcellán, Darboux transformation and perturbation of linear functionals, Linear Algebra Appl. 384 (2004), 215–242. MR 2055354, DOI 10.1016/j.laa.2004.02.004
T. S. Chihara, On kernel polynomials and related systems, Boll. Un. Mat. Ital. (3) 19 (1964), 451–459. MR 0176121
S. Corteel, R. Stanley, D. Stanton, and L. Williams, Formulae for Askey-Wilson moments and enumeration of staircase tableaux, Trans. Amer. Math. Soc. 364 (2012), no. 11, 6009–6037. MR 2946941, DOI 10.1090/S0002-9947-2012-05588-7
Philippe Delsarte and Yves V. Genin, The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process. 34 (1986), no. 3, 470–478. MR 844658, DOI 10.1109/TASSP.1986.1164830
Maxim Derevyagin, Generalized Jacobi operators in Krein spaces, J. Math. Anal. Appl. 349 (2009), no. 2, 568–582. MR 2456213, DOI 10.1016/j.jmaa.2008.09.032
Maxim Derevyagin and Vladimir Derkach, Darboux transformations of Jacobi matrices and Padé approximation, Linear Algebra Appl. 435 (2011), no. 12, 3056–3084. MR 2831597, DOI 10.1016/j.laa.2011.05.035
Maxim Derevyagin, Luc Vinet, and Alexei Zhedanov, CMV matrices and little and big $-1$ Jacobi polynomials, Constr. Approx. 36 (2012), no. 3, 513–535. MR 2996442, DOI 10.1007/s00365-012-9164-0
Maxim S. Derevyagin and Alexei S. Zhedanov, An operator approach to multipoint Padé approximations, J. Approx. Theory 157 (2009), no. 1, 70–88. MR 2500154, DOI 10.1016/j.jat.2008.07.002
Fabian H. L. Essler and Vladimir Rittenberg, Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, J. Phys. A 29 (1996), no. 13, 3375–3407. MR 1400161, DOI 10.1088/0305-4470/29/13/013
Graham M. L. Gladwell, Inverse problems in vibration, 2nd ed., Solid Mechanics and its Applications, vol. 119, Kluwer Academic Publishers, Dordrecht, 2004. MR 2102477
F. Alberto Grünbaum and Luc Haine, Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation, Symmetries and integrability of difference equations (Estérel, PQ, 1994) CRM Proc. Lecture Notes, vol. 9, Amer. Math. Soc., Providence, RI, 1996, pp. 143–154. MR 1416834, DOI 10.1090/crmp/009/14
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
Vasyl L. Ostrovskyĭ and Yuriĭ S. Samoĭlenko, On pairs of selfadjoint operators, Sem. Sophus Lie 3 (1993), no. 2, 185–218. MR 1270175
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
B. D. Sleeman, Multiparameter spectral theory in Hilbert space, J. Math. Anal. Appl. 65 (1978), no. 3, 511–530. MR 510467, DOI 10.1016/0022-247X(78)90160-9
Vyacheslav Spiridonov, Luc Vinet, and Alexei Zhedanov, Spectral transformations, self-similar reductions and orthogonal polynomials, J. Phys. A 30 (1997), no. 21, 7621–7637. MR 1603364, DOI 10.1088/0305-4470/30/21/030
Vyacheslav Spiridonov and Alexei Zhedanov, Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials, Methods Appl. Anal. 2 (1995), no. 4, 369–398. MR 1376302, DOI 10.4310/MAA.1995.v2.n4.a1
Vyacheslav Spiridonov and Alexei Zhedanov, Discrete-time Volterra chain and classical orthogonal polynomials, J. Phys. A 30 (1997), no. 24, 8727–8737. MR 1619538, DOI 10.1088/0305-4470/30/24/031
Wim Schoutens and Jozef L. Teugels, Lévy processes, polynomials and martingales, Comm. Statist. Stochastic Models 14 (1998), no. 1-2, 335–349. Special issue in honor of Marcel F. Neuts. MR 1617536, DOI 10.1080/15326349808807475
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Dunkl shift operators and Bannai-Ito polynomials, Adv. Math. 229 (2012), no. 4, 2123–2158. MR 2880217, DOI 10.1016/j.aim.2011.12.020
Masaru Uchiyama, Tomohiro Sasamoto, and Miki Wadati, Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials, J. Phys. A 37 (2004), no. 18, 4985–5002. MR 2065218, DOI 10.1088/0305-4470/37/18/006
A. P. Veselov and A. B. Shabat, A dressing chain and the spectral theory of the Schrödinger operator, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 1–21, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 2, 81–96. MR 1251164, DOI 10.1007/BF01085979
Luc Vinet and Alexei Zhedanov, Szegö polynomials on the real axis, Integral Transform. Spec. Funct. 8 (1999), no. 1-2, 149–164. MR 1730600, DOI 10.1080/10652469908819223
Luc Vinet and Alexei Zhedanov, A ‘missing’ family of classical orthogonal polynomials, J. Phys. A 44 (2011), no. 8, 085201, 16. MR 2770369, DOI 10.1088/1751-8113/44/8/085201
Luc Vinet and Alexei Zhedanov, A limit $q=-1$ for the big $q$-Jacobi polynomials, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5491–5507. MR 2931336, DOI 10.1090/S0002-9947-2012-05539-5
Luc Vinet and Alexei Zhedanov, A Bochner theorem for Dunkl polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 020, 9. MR 2804576, DOI 10.3842/SIGMA.2011.020
David S. Watkins, Isospectral flows, SIAM Rev. 26 (1984), no. 3, 379–391. MR 750456, DOI 10.1137/1026075
David S. Watkins, Some perspectives on the eigenvalue problem, SIAM Rev. 35 (1993), no. 3, 430–471. MR 1234638, DOI 10.1137/1035090
D. S. Watkins and L. Elsner, Self-equivalent flows associated with the generalized eigenvalue problem, Linear Algebra Appl. 118 (1989), 107–127. MR 995370, DOI 10.1016/0024-3795(89)90576-4
Alexei Zhedanov, On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval, J. Approx. Theory 94 (1998), no. 1, 73–106. MR 1637803, DOI 10.1006/jath.1998.3179
Alexei Zhedanov, Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 (1997), no. 1, 67–86. MR 1482157, DOI 10.1016/S0377-0427(97)00130-1
Alexei Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory 101 (1999), no. 2, 303–329. MR 1726460, DOI 10.1006/jath.1999.3339