A “Continuous” Limit of the Complementary Bannai–Ito Polynomials: Chihara Polynomials

Vincent X. Genest ; Luc Vinet ; Alexei Zhedanov

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

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.

Détail

Keywords: Bannai–Ito polynomials; Dunkl operators; orthogonal polynomials; quadratic algebras.

Bibliographie

[1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize {J}acobi polynomials, Mem. Amer. Math. Soc., 54, no. 319, 1985, iv+55 pp. | DOI | MR

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

[3] Belmehdi S., “Generalized {G}egenbauer orthogonal polynomials”, J. Comput. Appl. Math., 133 (2001), 195–205 | DOI | MR | Zbl

[4] Ben Cheikh Y., Gaied M., “Characterization of the {D}unkl-classical symmetric orthogonal polynomials”, Appl. Math. Comput., 187 (2007), 105–114 | DOI | MR | Zbl

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

[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, 81, Cambridge University Press, Cambridge, 2001 | DOI | MR | Zbl

[8] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[9] Genest V. X., Ismail M. E. H., Vinet L., Zhedanov A., “The {D}unkl oscillator in the plane. I: {S}uperintegrability, separated wavefunctions and overlap coefficients”, J. Phys. A: Math. Theor., 46 (2013), 145201, 21 pp., arXiv: 1212.4459 | DOI | MR | Zbl

[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 | DOI

[11] Genest V. X., Vinet L., Zhedanov A., “Bispectrality of the complementary {B}annai–{I}to polynomials”, SIGMA, 9 (2013), 018, 20 pp., arXiv: 1211.2461 | DOI | MR | Zbl

[12] Genest V. X., Vinet L., Zhedanov A., “The singular and the {$2:1$} anisotropic {D}unkl oscillators in the plane”, J. Phys. A: Math. Theor., 46 (2013), 325201, 17 pp., arXiv: 1305.2126 | DOI | MR | Zbl

[13] Genest V. X., Vinet L., Zhedanov A., “The {B}annai–{I}to polynomials as {R}acah coefficients of the {$sl_{-1}(2)$} algebra”, Proc. Amer. Math. Soc., 142 (2014), 1545–1560, arXiv: 1205.4215 | DOI | MR | Zbl

[14] Geronimus J., “The orthogonality of some systems of polynomials”, Duke Math. J., 14 (1947), 503–510 | DOI | MR | Zbl

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

[16] 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

[17] Koornwinder T. H., “On the limit from {$q$}-{R}acah polynomials to big {$q$}-{J}acobi polynomials”, SIGMA, 7 (2011), 040, 8 pp., arXiv: 1011.5585 | DOI | MR | Zbl

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

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

[20] Rosenblum M., “Generalized {H}ermite polynomials and the {B}ose-like oscillator calculus”, Nonselfadjoint Operators and Related Topics ({B}eer {S}heva, 1992), Oper. Theory Adv. Appl., 73, Birkhäuser, Basel, 1994, 369–396 | MR | Zbl

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

[22] 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

[23] Vinet L., Zhedanov A., “A {B}ochner theorem for {D}unkl polynomials”, SIGMA, 7 (2011), 020, 9 pp., arXiv: 1011.1457 | DOI | MR | Zbl

[24] 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

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