Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 038, 18 pages      arXiv:1309.7235      https://doi.org/10.3842/SIGMA.2014.038

A ''Continuous'' Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials

Vincent X. Genest a, Luc Vinet a and Alexei Zhedanov b
a) Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, QC, Canada, H3C 3J7
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received December 23, 2013, in final form March 24, 2014; Published online March 30, 2014

Abstract
A novel family of $-1$ orthogonal polynomials called the Chihara polynomials is characterized. The polynomials are obtained from a ''continuous'' limit of the complementary Bannai-Ito polynomials, which are the kernel partners of the Bannai-Ito polynomials. The three-term recurrence relation and the explicit expression in terms of Gauss hypergeometric functions are obtained through a limit process. A one-parameter family of second-order differential Dunkl operators having these polynomials as eigenfunctions is also exhibited. The quadratic algebra with involution encoding this bispectrality is obtained. The orthogonality measure is derived in two different ways: by using Chihara's method for kernel polynomials and, by obtaining the symmetry factor for the one-parameter family of Dunkl operators. It is shown that the polynomials are related to the big $-1$ Jacobi polynomials by a Christoffel transformation and that they can be obtained from the big $q$-Jacobi by a $q\rightarrow -1$ limit. The generalized Gegenbauer/Hermite polynomials are respectively seen to be special/limiting cases of the Chihara polynomials. A one-parameter extension of the generalized Hermite polynomials is proposed.

Key words: Bannai-Ito polynomials; Dunkl operators; orthogonal polynomials; quadratic algebras.

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References

  1. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55 pages.
  2. Bannai E., Ito T., Algebraic combinatorics. I. Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984.
  3. Belmehdi S., Generalized Gegenbauer orthogonal polynomials, J. Comput. Appl. Math. 133 (2001), 195-205.
  4. Ben Cheikh Y., Gaied M., Characterization of the Dunkl-classical symmetric orthogonal polynomials, Appl. Math. Comput. 187 (2007), 105-114.
  5. Chihara T.S., On kernel polynomials and related systems, Boll. Un. Mat. Ital. 19 (1964), 451-459.
  6. Chihara T.S., An introduction to orthogonal polynomials, Dover Books on Mathematics, Dover Publications, New York, 2011.
  7. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  8. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
  9. Genest V.X., Ismail M.E.H., Vinet L., Zhedanov A., The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients, J. Phys. A: Math. Theor. 46 (2013), 145201, 21 pages, arXiv:1212.4459.
  10. Genest V.X., Ismail M.E.H., Vinet L., Zhedanov A., The Dunkl oscillator in the plane: II. Representations of the symmetry algebra, Comm. Math. Phys., to appear, arXiv:1302.6142.
  11. Genest V.X., Vinet L., Zhedanov A., Bispectrality of the complementary Bannai-Ito polynomials, SIGMA 9 (2013), 018, 20 pages, arXiv:1211.2461.
  12. Genest V.X., Vinet L., Zhedanov A., The singular and the $2:1$ anisotropic Dunkl oscillators in the plane, J. Phys. A: Math. Theor. 46 (2013), 325201, 17 pages, arXiv:1305.2126.
  13. Genest V.X., Vinet L., Zhedanov A., The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra, Proc. Amer. Math. Soc. 142 (2014), 1545-1560, arXiv:1205.4215.
  14. Geronimus J., The orthogonality of some systems of polynomials, Duke Math. J. 14 (1947), 503-510.
  15. Granovskii Y.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
  16. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  17. Koornwinder T.H., On the limit from $q$-Racah polynomials to big $q$-Jacobi polynomials, SIGMA 7 (2011), 040, 8 pages, arXiv:1011.5585.
  18. Leonard D.A., Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), 656-663.
  19. Marcellán F., Petronilho J., Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl. 260 (1997), 169-208.
  20. Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel, 1994, 369-396.
  21. Tsujimoto S., Vinet L., Zhedanov A., Dunkl shift operators and Bannai-Ito polynomials, Adv. Math. 229 (2012), 2123-2158, arXiv:1106.3512.
  22. Tsujimoto S., Vinet L., Zhedanov A., Dual $-1$ Hahn polynomials: "classical" polynomials beyond the Leonard duality, Proc. Amer. Math. Soc. 141 (2013), 959-970, arXiv:1108.0132.
  23. Vinet L., Zhedanov A., A Bochner theorem for Dunkl polynomials, SIGMA 7 (2011), 020, 9 pages, arXiv:1011.1457.
  24. Vinet L., Zhedanov A., A 'missing' family of classical orthogonal polynomials, J. Phys. A: Math. Theor. 44 (2011), 085201, 16 pages, arXiv:1011.1669.
  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.

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