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

SIGMA 7 (2011), 020, 9 pages      arXiv:1011.1457
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

A Bochner Theorem for Dunkl Polynomials

Luc Vinet a and Alexei Zhedanov b
a) Centre de recherches mathématiques Universite de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 Canada
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received November 30, 2010, in final form February 25, 2011; Published online February 27, 2011

We establish an analogue of the Bochner theorem for first order operators of Dunkl type, that is we classify all such operators having polynomial solutions. Under natural conditions it is seen that the only families of orthogonal polynomials in this category are limits of little and big q-Jacobi polynomials as q=−1.

Key words: classical orthogonal polynomials; Dunkl operators; Jacobi polynomials; little q-Jacobi polynomials; big q-Jacobi polynomials.

pdf (304 kb)   tex (12 kb)


  1. Bannai E., Ito T., Algebraic combinatorics. I. Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984.
  2. Belmehdi S., Generalized Gegenbauer orthogonal polynomials, J. Comput. Appl. Math. 133 (2001), 195-205.
  3. Ben Cheikh Y., Gaied M., Characterization of the Dunkl-classical symmetric orthogonal polynomials, Appl. Math. Comput. 187, (2007), 105-114.
  4. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  5. Chouchene F., Harmonic analysis associated with the Jacobi-Dunkl operator on ]−π/2,π/2[, J. Comput. Appl. Math. 178, (2005), 75-89.
  6. Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227.
  7. Everitt W.N., Kwon K.H., Littlejohn L.L., Wellman R., Orthogonal polynomial solutions of linear ordinary differential equations, J. Comput. Appl. Math. 133 (2001), 85-109.
  8. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  9. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  10. Koornwinder T., Bouzeffour F., Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials, Appl. Anal., to appear, arXiv:1006.1140.
  11. Littlejohn L.L., Race D., Symmetric and symmetrisable differential expressions, Proc. London Math. Soc. 60 (1990), 344-364.
  12. Nevai P., Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213.
  13. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
  14. 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, math.CA/9307224.
  15. Vinet L., Zhedanov A., A "missing" family of classical orthogonal polynomials, J. Phys. A: Math. Theor. 44 (2011), 085201, 16 pages, arXiv:1011.1669.
  16. Vinet L., Zhedanov A., A limit q=−1 for big q-Jacobi polynomials, Trans. Amer. Math. Soc., to appear, arXiv:1011.1429.

Previous article   Next article   Contents of Volume 7 (2011)