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


SIGMA 7 (2011), 092, 20 pages      arXiv:1110.0580      https://doi.org/10.3842/SIGMA.2011.092
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

An Introduction to the q-Laguerre-Hahn Orthogonal q-Polynomials

Abdallah Ghressi, Lotfi Khériji and Mohamed Ihsen Tounsi
Institut Supérieur des Sciences Appliquées et de Technologies de Gabès, Rue Omar Ibn El-Khattab 6072 Gabès, Tunisia

Received February 14, 2011, in final form September 26, 2011; Published online October 04, 2011

Abstract
Orthogonal q-polynomials associated with q-Laguerre-Hahn form will be studied as a generalization of the q-semiclassical forms via a suitable q-difference equation. The concept of class and a criterion to determinate it will be given. The q-Riccati equation satisfied by the corresponding formal Stieltjes series is obtained. Also, the structure relation is established. Some illustrative examples are highlighted.

Key words: orthogonal q-polynomials; q-Laguerre-Hahn form; q-difference operator; q-difference equation; q-Riccati equation.

pdf (417 kb)   tex (20 kb)

References

  1. Alaya J., Maroni P., Symmetric Laguerre-Hahn forms of class s=1, Integral Transform. Spec. Funct. 2 (1996), 301-320.
  2. Alaya J., Maroni P., Some semi-classical and Laguerre-Hahn forms defined by pseudo-functions, Methods Appl. Anal. 3 (1996), 12-30.
  3. Álvarez-Nodarse R., Medem J.C., q-classical polynomials and the q-Askey and Nikiforov-Uvarov tableaux, J. Comput. Appl. Math. 135 (2001), 197-223.
  4. Bangerezako G., The fourth order difference equation for the Laguerre-Hahn polynomials orthogonal on special non-uniform lattices, Ramanujan J. 5 (2001), 167-181.
  5. Bangerezako G., An introduction to q-difference equations, Bujumbura, 2008.
  6. Bouakkaz H., Maroni P., Description des polynômes de Laguerre-Hahn de classe zéro, in Orthogonal Polynomials and Their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., Vol. 9, Baltzer, Basel, 1991, 189-194.
  7. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  8. Dini J., Sur les formes linéaires et polynômes oerthogonaux de Laguerre-Hahn, Thèse de Doctorat, Université Pierre et Marie Curie, Paris VI, 1988.
  9. Dini J., Maroni P., Ronveaux A., Sur une perturbation de la récurrence vérifiée par une suite de polynômes orthogonaux, Portugal. Math. 46 (1989), 269-282.
  10. Dzoumba J., Sur les polynômes de Laguerre-Hahn, Thèse de 3 ème cycle, Université Pierre et Marie Curie, Paris VI, 1985.
  11. Foupouagnigni M., Ronveaux A., Koepf W., Fourth order q-difference equation for the first associated of the q-classical orthogonal polynomials, J. Comput. Appl. Math. 101 (1999), 231-236.
  12. Foupouagnigni M., Ronveaux A., Difference equation for the co-recursive rth associated orthogonal polynomials of the Dq-Laguerre-Hahn class, J. Comput. Appl. Math. 153 (2003), 213-223.
  13. Foupouagnigni M., Marcellán F., Characterization of the Dϖ-Laguerre-Hahn functionals, J. Difference Equ. Appl. 8 (2002), 689-717.
  14. Ghressi A., Khériji L., The symmetrical Hq-semiclassical orthogonal polynomials of class one, SIGMA 5 (2009), 076, 22 pages, arXiv:0907.3851.
  15. Guerfi M., Les polynômes de Laguerre-Hahn affines discrets, Thèse de troisième cycle, Univ. P. et M. Curie, Paris, 1988.
  16. Khériji L., Maroni P., The Hq-classical orthogonal polynomials, Acta. Appl. Math. 71 (2002), 49-115.
  17. Khériji L., An introduction to the Hq-semiclassical orthogonal polynomials, Methods Appl. Anal. 10 (2003), 387-411.
  18. Magnus A., Riccati acceleration of Jacobi continued fractions and Laguerre-Hahn orthogonal polynomials, in Padé Approximation and its Applications (Bad Honnef, 1983), Lecture Notes in Math., Vol. 1071, Springer, Berlin, 1984, 213-230.
  19. Marcellán F., Salto M., Discrete semiclassical orthogonal polynomials, J. Difference. Equ. Appl. 4 (1998), 463-496.
  20. Maroni P., Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classique, in Orthogonal Polynomials and their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., Vol. 9, Baltzer, Basel, 1991, 95-130.
  21. Medem J.C., Álvarez-Nodarse R., Marcellán F., On the q-polynomials: a distributional study, J. Comput. Appl. Math. 135 (2001), 157-196.


Previous article   Next article   Contents of Volume 7 (2011)