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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 54263, 15 pages
http://dx.doi.org/10.1155/IJMMS/2006/54263

Linearization coefficients for Sheffer polynomial sets via lowering operators

1Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, Monastir 5019, Tunisia
2Département de Préparation en Math-Physique, Institut Préparatoire aux Études d'Ingénieur de Monastir, Monastir 5019, Tunisia

Received 16 May 2005; Revised 2 March 2006; Accepted 12 March 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Álvarez-Nodarse, R. J. Yáñez, and J. S. Dehesa, “Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials,” Journal of Computational and Applied Mathematics, vol. 89, no. 1, pp. 171–197, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. Appell, “Sur une classe de polynômes,” Annales Scientifiques de l'École Normale Supérieure, vol. 9, no. 2, pp. 119–144, 1880. View at Google Scholar · View at MathSciNet
  3. T. K. Araaya, “Linearization and connection problems for the symmetric Meixner-Pollaczek polynomials,” Uppsala University (2003), 59–70.
  4. I. Area, E. Godoy, A. Ronveaux, and A. Zarzo, “Solving connection and linearization problems within the Askey scheme and its q-analogue via inversion formulas,” Journal of Computational and Applied Mathematics, vol. 133, no. 1-2, pp. 151–162, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. P. L. Artés, J. S. Dehesa, A. Martínez-Finkelshtein, and J. Sánchez-Ruiz, “Linearization and connection coefficients for hypergeometric-type polynomials,” Journal of Computational and Applied Mathematics, vol. 99, no. 1-2, pp. 15–26, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. R. Askey, “Orthogonal polynomials and positivity,” in Studies in Applied Mathematics, vol. 6 of Special Functions and Wave Propagation, pp. 64–85, SIAM, Pennsylvania, 1970. View at Google Scholar
  7. R. Askey, Orthogonal Polynomials and Special Functions, vol. 21 of CBMS Regional Conference Series, SIAM, Pennsylvania, 1975. View at MathSciNet
  8. R. Askey and G. Gasper, “Linearization of the product of Jacobi polynomials. III,” Canadian Journal of Mathematics, vol. 23, pp. 332–338, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. Askey and G. Gasper, “Convolution structures for Laguerre polynomials,” Journal d'Analyse Mathématique, vol. 31, pp. 48–68, 1977. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Belmehdi, S. Lewanowicz, and A. Ronveaux, “Linearization of the product of orthogonal polynomials of a discrete variable,” Applicationes Mathematicae, vol. 24, no. 4, pp. 445–455, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Ben Cheikh, “On obtaining dual sequences via quasi-monomiality,” Georgian Mathematical Journal, vol. 9, no. 3, pp. 413–422, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Y. Ben Cheikh, “Some results on quasi-monomiality,” Applied Mathematics and Computation, vol. 141, no. 1, pp. 63–76, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. Ben Cheikh and H. Chaggara, “Connection problems via lowering operators,” Journal of Computational and Applied Mathematics, vol. 178, no. 1-2, pp. 45–61, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Ben Cheikh and H. Chaggara, “Connection coefficients between Boas-Buck polynomial sets,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 665–689, 2006. View at Publisher · View at Google Scholar
  15. L. Carlitz, “The product of certain polynomials analogous to the Hermite polynomials,” The American Mathematical Monthly, vol. 64, pp. 723–725, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. L. Carlitz, “Products of Appell polynomials,” Collectanea Mathematica, vol. 15, pp. 245–258, 1963. View at Google Scholar · View at Zentralblatt MATH
  17. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. View at Zentralblatt MATH · View at MathSciNet
  18. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vols. I, II, McGraw-Hill, New York, 1953. View at MathSciNet
  19. G. Gasper, “Linearization of the product of Jacobi polynomials. I,” Canadian Journal of Mathematics, vol. 22, pp. 171–175, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. C. Jordan, Calculus of Finite Differences, Chelsea, New York, 2nd edition, 1960.
  21. D. Kim and J. Zeng, “A combinatorial formula for the linearization coefficients of general Sheffer polynomials,” European Journal of Combinatorics, vol. 22, no. 3, pp. 313–332, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H. Kleindienst and A. Lüchow, “Multiplication theorems for orthogonal polynomials,” International Journal of Quantum Chemistry, vol. 48, no. 4, pp. 239–247, 1993. View at Publisher · View at Google Scholar
  23. R. Koekoek and R. F. Swarrtow, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,” Tech. Rep. 98–17, Faculty of the Technical Mathematics and Informatics, Delft University of Technology, Delft, 1998. View at Google Scholar
  24. C. Markett, “Linearization of the product of symmetric orthogonal polynomials,” Constructive Approximation, vol. 10, no. 3, pp. 317–338, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. E. D. Rainville, Special Functions, Macmillan, New York, 1960. View at Zentralblatt MATH · View at MathSciNet
  26. A. Ronveaux, “Orthogonal polynomials: connection and linearization coefficients,” in Proceedings of the International Workshop on Orthogonal Polynomials in Mathematical Physics, M. Alfano, R. Álvarez-Nodarse, G. López Lagomasino, and F. Marcellán, Eds., Madrid, June 1996. View at Google Scholar
  27. A. Ronveaux, M. N. Hounkonnou, and S. Belmehdi, “Generalized linearization problems,” Journal of Physics. A: Mathematical and General, vol. 28, no. 15, pp. 4423–4430, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. J. Sánchez-Ruiz, P. L. Artés, A. Martínez-Finkelshtein, and J. S. Dehesa, “Linearization problems of hypergeometric polynomials in quantum physics,” in Proceedings of the Melfi Workshop on Advanced Special Functions and Applications, G. Dattoli, H. M. Srivastava, and D. Cocolicchio, Eds., Rome, May 1999. View at Google Scholar
  29. J. Sánchez-Ruiz and J. S. Dehesa, “Some connection and linearization problems for polynomials in and beyond the Askey scheme,” Journal of Computational and Applied Mathematics, vol. 133, no. 1-2, pp. 579–591, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. I. M. Sheffer, “Some properties of polynomial sets of type zero,” Duke Mathematical Journal, vol. 5, no. 3, pp. 590–622, 1939. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. H. M. Srivastava, “On the reducibility of Appell's function F4,” Canadian Mathematical Bulletin, vol. 16, pp. 295–298, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. H. M. Srivastava, “A unified theory of polynomial expansions and their applications involving Clebsch-Gordan type linearization relations and Neumann series,” Astrophysics and Space Science, vol. 150, no. 2, pp. 251–266, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. H. M. Srivastava, “Some Clebesch-Gordan type linearization relations and other polynomial expansions associated with a class of generalized multiple hypergeometric series arising in physical and quantum chemical applications,” Journal of Physics A: Mathematical and General, vol. 21, pp. 4463–4470, 1988. View at Publisher · View at Google Scholar
  34. H. M. Srivastava and Y. Ben Cheikh, “Orthogonality of some polynomial sets via quasi-monomiality,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 415–425, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, John Willey & Sons, New York; Brisbane, Toronto, 1984. View at Zentralblatt MATH
  36. R. Szwarc, “Convolution structures associated with orthogonal polynomials,” Journal of Mathematical Analysis and Applications, vol. 170, no. 1, pp. 158–170, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. R. Szwarc, “Nonnegative linearization and quadratic transformation of Askey-Wilson polynomials,” Canadian Mathematical Bulletin, vol. 39, no. 2, pp. 241–249, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. R. Szwarc, “A necessary and sufficient condition for nonnegative product linearization of orthogonal polynomials,” Constructive Approximation, vol. 19, no. 4, pp. 565–573, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. J. Zeng, “Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials,” Proceedings of the London Mathematical Society. Third Series, vol. 65, no. 1, pp. 1–22, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet