Geodesic
Bispectrality of the Complementary Bannai–Ito Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A one-parameter family of operators that have the complementary Bannai–Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai–Ito polynomials and also correspond to a $q\rightarrow-1$ limit of the Askey–Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey–Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual $-1$ Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.
Keywords: Bannai–Ito polynomials; quadratic algebras; Dunkl operators. 
@article{SIGMA_2013_9_a17,
     author = {Vincent X. Genest and Luc Vinet and Alexei Zhedanov},
     title = {Bispectrality of the {Complementary} {Bannai{\textendash}Ito} {Polynomials}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a17/}
}
TY  - JOUR
AU  - Vincent X. Genest
AU  - Luc Vinet
AU  - Alexei Zhedanov
TI  - Bispectrality of the Complementary Bannai–Ito Polynomials
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2013
VL  - 9
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a17/
LA  - en
ID  - SIGMA_2013_9_a17
ER  - 
%0 Journal Article
%A Vincent X. Genest
%A Luc Vinet
%A Alexei Zhedanov
%T Bispectrality of the Complementary Bannai–Ito Polynomials
%J Symmetry, integrability and geometry: methods and applications
%D 2013
%V 9
%U http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a17/
%G en
%F SIGMA_2013_9_a17
Vincent X. Genest; Luc Vinet; Alexei Zhedanov. Bispectrality of the Complementary Bannai–Ito Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a17/

[1] Bannai E., Ito T., Algebraic combinatorics, v. I, Association schemes, The Benjamin/Cummings Publishing Co. Inc., Menlo Park, CA, 1984 | MR

[2] Chihara T. S., An introduction to orthogonal polynomials, Mathematics and its Applications, 13, Gordon and Breach Science Publishers, New York, 1978 | MR | Zbl

[3] Chihara T. S., “On kernel polynomials and related systems”, Boll. Un. Mat. Ital. (3), 19 (1964), 451–459 | MR | Zbl

[4] Dunkl C. F., “Integral kernels with reflection group invariance”, Canad. J. Math., 43 (1991), 1213–1227 | DOI | MR | Zbl

[5] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[6] Genest V. X., Vinet L., Zhedanov A., “The algebra of dual $-1$ {H}ahn polynomials and the {C}lebsch–{G}ordan problem of $sl_{-1}(2)$”, J. Math. Phys., 54 (2013), 023506, 13 pp., arXiv: 1207.4220 | DOI

[7] Genest V. X., Vinet L., Zhedanov A., “The Bannai–Ito polynomials as {R}acah coefficients of the $sl_{-1}(2)$ algebra”, Proc. Amer. Math. Soc. (to appear) , arXiv: 1205.4215

[8] Granovskiĭ Y. I., Lutzenko I. M., Zhedanov A. S., “Mutual integrability, quadratic algebras, and dynamical symmetry”, Ann. Physics, 217 (1992), 1–20 | DOI | MR | Zbl

[9] Jafarov E. I., Stoilova N. I., Van der Jeugt J., “Finite oscillator models: the Hahn oscillator”, J. Phys. A: Math. Theor., 44 (2011), 265203, 15 pp., arXiv: 1101.5310 | DOI | Zbl

[10] Jafarov E. I., Stoilova N. I., Van der Jeugt J., “The $\mathfrak{su}(2)_\alpha$ Hahn oscillator and a discrete Fourier–Hahn transform”, J. Phys. A: Math. Theor., 44 (2011), 355205, 18 pp., arXiv: 1106.1083 | DOI | Zbl

[11] Karlin S., McGregor J. L., “The differential equations of birth-and-death processes, and the Stieltjes moment problem”, Trans. Amer. Math. Soc., 85 (1957), 489–546 | DOI | MR | Zbl

[12] Karlin S., McGregor J. L., “The Hahn polynomials, formulas and an application”, Scripta Math., 26 (1961), 33–46 | MR | Zbl

[13] Koekoek R., Lesky P. A., Swarttouw R. F., Hypergeometric orthogonal polynomials and their {$q$}-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[14] Leonard D. A., “Orthogonal polynomials, duality and association schemes”, SIAM J. Math. Anal., 13 (1982), 656–663 | DOI | MR | Zbl

[15] Marcellán F., Petronilho J., “Eigenproblems for tridiagonal $2$-Toeplitz matrices and quadratic polynomial mappings”, Linear Algebra Appl., 260 (1997), 169–208 | DOI | MR | Zbl

[16] Szeg{ö} G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975

[17] Terwilliger P., “The universal Askey–Wilson algebra and the equitable presentation of $U_q(sl_2)$”, SIGMA, 7 (2011), 099, 26 pp., arXiv: 1107.3544 | DOI | MR | Zbl

[18] Terwilliger P., “Two linear transformations each tridiagonal with respect to an eigenbasis of the other”, Linear Algebra Appl., 330 (2001), 149–203, arXiv: math.RA/0406555 | DOI | MR | Zbl

[19] Terwilliger P., Vidunas R., “Leonard pairs and the Askey–Wilson relations”, J. Algebra Appl., 3 (2004), 411–426, arXiv: math.QA/0305356 | DOI | MR | Zbl

[20] Tsujimoto S., Vinet L., Zhedanov A., “Dual {$-1$} {H}ahn polynomials: “classical” polynomials beyond the {L}eonard duality”, Proc. Amer. Math. Soc., 141 (2013), 959–970, arXiv: 1108.0132 | DOI | MR | Zbl

[21] Tsujimoto S., Vinet L., Zhedanov A., “Dunkl shift operators and Bannai–Ito polynomials”, Adv. Math., 229 (2012), 2123–2158, arXiv: 1106.3512 | DOI | MR | Zbl

[22] Tsujimoto S., Vinet L., Zhedanov A., “From $sl_q(2)$ to a parabosonic Hopf algebra”, SIGMA, 7 (2011), 093, 13 pp., arXiv: 1108.1603 | DOI | MR | Zbl

[23] Vidunas R., “Askey–Wilson relations and Leonard pairs”, Discrete Math., 308 (2008), 479–495, arXiv: math.QA/0511509 | DOI | MR | Zbl

[24] Vinet L., Zhedanov A., “A Bochner theorem for Dunkl polynomials”, SIGMA, 7 (2011), 020, 9 pp., arXiv: 1011.1457 | DOI | MR | Zbl

[25] Vinet L., Zhedanov A., “A limit $q=-1$ for the big $q$-Jacobi polynomials”, Trans. Amer. Math. Soc., 364 (2012), 5491–5507, arXiv: 1011.1429 | DOI | MR

[26] Vinet L., Zhedanov A., “A ‘missing’ family of classical orthogonal polynomials”, J. Phys. A: Math. Theor., 44 (2011), 085201, 16 pp., arXiv: 1011.1669 | DOI | MR | Zbl

[27] Vinet L., Zhedanov A., “Dual $-1$ Hahn polynomials and perfect state transfer”, J. Phys. Conf. Ser., 343 (2012), 012125, 10 pp., arXiv: 1110.6477 | DOI | MR

[28] Vinet L., Zhedanov A., “Para-Krawtchouk polynomials on a bi-lattice and a quantum spin chain with perfect state transfer”, J. Phys. A: Math. Theor., 45 (2012), 265304, 11 pp., arXiv: 1110.6475 | DOI | MR | Zbl

[29] Zhedanov A., ““{H}idden symmetry” of Askey–Wilson polynomials”, Theoret. and Math. Phys., 89 (1991), 1146–1157 | DOI | MR | Zbl

[30] Zhedanov A., “Rational spectral transformations and orthogonal polynomials”, J. Comput. Appl. Math., 85 (1997), 67–86 | DOI | MR | Zbl

[31] Zhedanov A., “The “{H}iggs algebra” as a “quantum” deformation of {${\rm SU}(2)$}”, Modern Phys. Lett. A, 7 (1992), 507–512 | DOI | MR | Zbl