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q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0 FOR POSITIVE INTEGERS N

  • Moreno, Samuel G. ;
  • Garcia-Caballe, Esther M.
  • Received : 2009.11.18
  • Published : 2011.09.01

Abstract

The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

Keywords

non-standard orthogonality;q-Laguerre polynomials;basic hypergeometric series

References

  1. M. Alvarez de Morales, T. E. Perez, M. A. Pinar, and A. Ronveaux, Non-standard orthogonality for Meixner polynomials, Electron. Trans. Numer. Anal. 9 (1999), 1-25.
  2. M. V. DeFazio, D. P. Gupta, and M. E. Muldoon, Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials, J. Math. Anal. Appl. 334 (2007), no. 2, 977-982. https://doi.org/10.1016/j.jmaa.2007.01.008
  3. G. Gasper and M. Rahman, Basic Hypergeometric Series, Second edition.Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004.
  4. W. Hahn, Uber orthogonalpolynome die q-differenzgleichingen genugen, Math. Nachr. 2 (1949), 4-34. https://doi.org/10.1002/mana.19490020103
  5. M. E. H. Ismail and M. Rahman, The q-Laguerre polynomials and related moment problems, J. Math. Anal. Appl. 218 (1998), no. 1, 155-174. https://doi.org/10.1006/jmaa.1997.5771
  6. K. H. Kwon and L. L. Littlejohn, The orthogonality of the Laguerre polynomials ${L-n^{-k}({\mathit{x}})}$ for positive integers k, Ann. Numer. Math. 2 (1995), no. 1-4, 289-303.
  7. J. Koekoek and R. Koekoek, A note on the q-derivative operator, J. Math. Anal. Appl. 176 (1993), no. 2, 627-634. https://doi.org/10.1006/jmaa.1993.1237
  8. R. Koekoek, Generalizations of a q-analogue of Laguerre polynomials, J. Approx. Theory 69 (1992), no. 1, 55-83. https://doi.org/10.1016/0021-9045(92)90049-T
  9. R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Technical Report 98-17, Delft University of Technology, 1998.
  10. D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), no. 1, 20-47. https://doi.org/10.1016/0022-247X(81)90048-2
  11. S. G. Moreno and E. M. Garcia-Caballero, Linear interpolation and Sobolev orthogonality, J. Approx. Theory 161 (2009), no. 1, 35-48. https://doi.org/10.1016/j.jat.2008.08.005
  12. S. G. Moreno and E. M. Garcia-Caballero, Non-standard orthogonality for the little q-Laguerre polynomials, Appl. Math. Lett. 22 (2009), no. 11, 1745-1749. https://doi.org/10.1016/j.aml.2009.05.017
  13. S. G. Moreno and E. M. Garcia-Caballero, Orthogonality of the Meixner-Pollaczek polynomials beyond Favard's theorem, submitted for publication.
  14. S. G. Moreno and E. M. Garcia-Caballero, Non-classical orthogonality relations for big and little q-Jacobi polynomials, J. Approx. Theory 162 (2010), no. 2, 303-322. https://doi.org/10.1016/j.jat.2009.05.008
  15. S. G. Moreno and E. M. Garcia-Caballero, Non-classical orthogonality relations for continuous q-Jacobi polynomials, Taiwan. J. Math, to appear.
  16. T. E. Perez and M. A. Pinar, On Sobolev orthogonality for the generalized Laguerre polynomials, J. Approx. Theory 86 (1996), no. 3, 278-285. https://doi.org/10.1006/jath.1996.0069
  17. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.

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