DOI QR코드

DOI QR Code

A DIFFERENTIAL EQUATION FOR MULTIPLE BESSEL POLYNOMIALS WITH RAISING AND LOWERING OPERATORS

  • Received : 2008.12.12
  • Published : 2011.05.31

Abstract

In this paper, we first find a raising operator and a lowering operator for multiple Bessel polynomials and then give a differential equation having multiple Bessel polynomials as solutions. Thus the differential equations were found for all multiple orthogonal polynomials that are orthogonal with respect to the same type of classical weights introduced by Aptekarev et al.

Keywords

multiple orthogonal polynomials;Bessel polynomials;multiple Bessel polynomial;differential equation

References

  1. A. I. Aptekarev, A. Branquinho, and W. Van Assche, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3887-3914. https://doi.org/10.1090/S0002-9947-03-03330-0
  2. B. Beckermann, J. Coussement, and W. Van Assche, Multiple Wilson and Jacobi-Pineiro polynomials, J. Approx. Theory 132 (2005), no. 2, 155-181. https://doi.org/10.1016/j.jat.2004.12.001
  3. M. G. de Bruin, Simultaneous Pade approximation and orthogonality, Orthogonal polynomials and applications (Bar-le-Duc, 1984), 74-83, Lecture Notes in Math., 1171, Springer, Berlin, 1985.
  4. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
  5. J. Coussement and W. Van Assche, Differential equations for multiple orthogonal polynomials with respect to classical weights: raising and lowering operators, J. Phys. A 39 (2006), no. 13, 3311-3318. https://doi.org/10.1088/0305-4470/39/13/010
  6. A. J. Duran, The Stieltjes moment problem for rapidly decreasing functions, Proc. Amer. Math. Soc. 107 (1989), no. 3, 731-741. https://doi.org/10.1090/S0002-9939-1989-0984787-0
  7. W. D. Evans, W. N. Everitt, K. H. Kwon, and L. L. Littlejohn, Real orthogonalizing weights for Bessel polynomials, J. Comput. Appl. Math. 49 (1993), no. 1-3, 51-57. https://doi.org/10.1016/0377-0427(93)90134-W
  8. A. M. Krall, Orthogonal polynomials through moment generating functionals, SIAM J. Math. Anal. 9 (1978), no. 4, 600-603. https://doi.org/10.1137/0509041
  9. H. L. Krall and O. Frink, A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), 100-115. https://doi.org/10.1090/S0002-9947-1949-0028473-1
  10. K. H. Kwon, S. S. Kim, and S. S. Han, Orthogonalizing weights of Tchebychev sets of polynomials, Bull. London Math. Soc. 24 (1992), no. 4, 361-367. https://doi.org/10.1112/blms/24.4.361
  11. K. H. Kwon, D. W. Lee, and L. L. Littlejohn, Differential equations having orthogonal polynomial solutions, J. Comput. Appl. Math. 80 (1997), no. 1, 1-16. https://doi.org/10.1016/S0377-0427(96)00096-9
  12. D. W. Lee, Properties of multiple Hermite and multiple Laguerre polynomials by the generating function, Integral Transforms Spec. Funct. 18 (2007), no. 11-12, 855-869. https://doi.org/10.1080/10652460701510725
  13. D. W. Lee, Difference equations for discrete classical multiple orthogonal polynomials, J. Approx. Theory 150 (2008), no. 2, 132-152. https://doi.org/10.1016/j.jat.2007.06.002
  14. R. D. Morton and A. M. Krall, Distributional weight functions for orthogonal polynomials, SIAM J. Math. Anal. 9 (1978), no. 4, 604-626. https://doi.org/10.1137/0509042
  15. E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs, Amer. Math. Soc., Providence, RI 92, 1991.
  16. G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc., Colloq. Publ. 213, Providence, RI, 1975.