Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED Δ-COHERENT PAIRS

  • Kwon, K.H. (Division of Applied Mathematic KAIST) ;
  • Lee, J.H. (Department of Mathematical Science SNU) ;
  • F. Marcellan (Departmento de Matematicas Universidad Carlos III de Madrid Avda)
  • Published : 2004.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

A pair of quasi-definite linear functionals {u$_{0}$, u$_1$} is a generalized $\Delta$-coherent pair if monic orthogonal polynomials (equation omitted) relative to u$_{0}$ and u$_1$, respectively, satisfy a relation (equation omitted) where $\sigma$$_{n}$ and T$_{n}$ are arbitrary constants and $\Delta$p = p($\chi$＋1) - p($\chi$) is the difference operator. We show that if {u$_{0}$, u$_1$} is a generalized $\Delta$-coherent pair, then u$_{0}$ and u$_{1}$ must be discrete-semiclassical linear functionals. We also find conditions under which either u$_{0}$ or u$_1$ is discrete-classical.ete-classical.

Keywords

References

  1. W. A. Al-Salam, Characterization Theorems for Orthogonal Polynomials, Orthogonal Polynomials: Theory and Practice (P. Nevai ed.), NATO Sci. Ser. C Math. Phys. Sci. 294 (1990), 1–24.
  2. I. Area, E. Godoy, and F. Marcellan, Classification of all $\Delta$-coherent pairs, Integral Transforms Spec. Funct. 9 (1) (2000), 1–18. https://doi.org/10.1080/10652460008819238
  3. I. Area, E. Godoy, and F. Marcellan, $\Delta$-coherent pairs and orthogonal polynomials of a discrete variable, Integral Transforms Spec. Funct. 13 (2002), 1–27. https://doi.org/10.1080/10652460304546
  4. M. G. de Bruin and H. G. Meijer, Zeros of orthogonal polynomials in a non-discrete Sobolev space, Ann. Numer. Math., 1995, 233–246.
  5. T. S. Chihara, On co-recursive orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957), 899–905. https://doi.org/10.1090/S0002-9939-1957-0092015-5
  6. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
  7. A. G. Garcia, F. Marcellan, and L. Salto, A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math. 57 (1995), 147–162. https://doi.org/10.1016/0377-0427(93)E0241-D
  8. A. Iserles, P. E. Koch, S. P. Norsett, and J. M. Sanz-Serna, On Polynomials Orthogonal with respect to Certain Sobolev Inner Products, J. Approx. Theory 65 (1991), 151–175. https://doi.org/10.1016/0021-9045(91)90100-O
  9. K. H. Kwon, D. W. Lee, F. Marcellan, and S. B. Park, On kernel polynomials and self perturbation of orthogonal polynomials, Ann. Mat. Pura Appl. (4) 180 (2) (2001), 127–146. https://doi.org/10.1007/s10231-001-8200-7
  10. K. H. Kwon, J. H. Lee, and F. Marcellan, Generalized coherent pairs, J. Math. Anal. Appl. 253 (2) (2001), 482–514. https://doi.org/10.1006/jmaa.2000.7157
  11. O. E. Lancaster, Orthogonal polynomials defined by difference equations, Amer. J. Math. 63 (1941), 185–207. https://doi.org/10.2307/2371289
  12. F. Marcellan, A. Martinez-Finkelshtein, and J. J. Moreno-Balcazar, k-coherence of measures with non-classical weights, In Margarita Mathematica to honnoure J. J. Guadalupe, L Espanol, and J. L. Varona Editors, Servicio Publicaciones Universidad de la Rioja, Logrono, 2001, 77–83.
  13. F. Marcellan, T. E. Perez, and M. A. Pinar, Orthogonal polynomials on weighted Sobolev spaces: the semiclassical case, Ann. Numer. Math. 2 (1995), 93–122.
  14. F. Marcellan and J. C. Petronilho, Orthogonal polynomials and coherent pairs: the classical case, Indag. Math. (N. S.) 6(3) (1995), 287–307. https://doi.org/10.1016/0019-3577(95)93197-I
  15. F. Marcellan, J. C. Petronilho, T. E. Perez, and M. A. Pinar, What is Beyond Coherent Pairs of Orthogonal Polynomials?, J. Comput. Appl. Math. 65 (1995), 267–277. https://doi.org/10.1016/0377-0427(95)00121-2
  16. F. Marcellan and L. Salto, Discrete semi-classical orthogonal polynomials, J. Difference Equ. Appl. 4 (1998), 463–496 https://doi.org/10.1080/10236199808808156
  17. H. G. Meijer, Coherent pairs and zeros of Sobolev-type orthogonal polynomials, Indag. Math. (N.S.) 4(2) (1993), 163–176. https://doi.org/10.1016/0019-3577(93)90037-Y
  18. H. G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (1997), 321–343. https://doi.org/10.1006/jath.1996.3062
  19. A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials Of A Discrete Variables, Springer-Verlag, Berlin, 1991.
  20. K. Pan, On Sobolev orthogonal polynomials with coherent pairs The Jacobi case, J. Comput. Appl. Math. 79 (1997), 249–262. https://doi.org/10.1016/S0377-0427(96)00168-9

Cited by

  1. A matrix characterization for theDν-semiclassical andDν-coherent orthogonal polynomials vol.487, 2015, https://doi.org/10.1016/j.laa.2015.09.014
  2. On linearly related sequences of difference derivatives of discrete orthogonal polynomials vol.284, 2015, https://doi.org/10.1016/j.cam.2014.06.018