Advertisement

Hypergeometric Multivariate Orthogonal Polynomials

  • Iván AreaEmail author
Conference paper
  • 40 Downloads
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

In this lecture a comparison between univariate and multivariate orthogonal polynomials is presented. The first step is to review classical univariate orthogonal polynomials, including classical continuous, classical discrete, their q-analogues and also classical orthogonal polynomials on nonuniform lattices. In all these cases, the orthogonal polynomials are solution of a second-order differential, difference, q-difference, or divided-difference equation of hypergeometric type. Next, a review multivariate orthogonal polynomials is presented. In the approach we have considered, the main tool is the partial differential, difference, q-difference or divided-difference equation of hypergeometric type the polynomial sequences satisfy. From these equations satisfied, the equation satisfied by any derivative (difference, q-difference or divided-difference) of the polynomials is obtained. A big difference appears for nonuniform lattices, where bivariate Racah and for bivariate q-Racah polynomials satisfy a fourth-order divided-difference equation of hypergeometric type. From this analysis, we propose a definition of multivariate classical orthogonal polynomials. Finally, some open problems are stated.

Keywords

Orthogonal polynomials Hypergeometric equation 

Mathematics Subject Classification (2000)

Primary 33C45 33C50; Secondary 39A13 39A14 

Notes

Acknowledgements

The authors thanks the reviewers for their careful reading of the manuscript. The author also wishes to thank Prof. Amílcar Branquinho for his valuable comments which improved a preliminary version of this manuscript.

The author thanks the financial support from the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016–75140–P, co-financed by the European Community fund FEDER.

References

  1. 1.
    W. Al-Salam, Characterization theorems for orthogonal polynomials, in Orthogonal Polynomials: Theory and Practice, ed. by P. Nevai (Kluwer Academic, Dordrecht, 1990), pp. 1–24Google Scholar
  2. 2.
    G.E. Andrews, R. Askey, Classical orthogonal polynomials, in Polynômes Orthogonaux et Applications, ed. by C. Brezinski et al. Lecture Notes in Mathematics, vol. 1171 (Springer, Berlin, 1985), pp. 36–62Google Scholar
  3. 3.
    P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques. Polynômes d’Hermite (Gauthier-Villars, Paris, 1926)Google Scholar
  4. 4.
    I. Area, Polinomios ortogonales de variable discreta: pares coherentes. Problemas de conexión. Ph.D. Thesis, Universidade de Vigo (1999)Google Scholar
  5. 5.
    I. Area, E. Godoy, On limit relations between some families of bivariate hypergeometric orthogonal polynomials. J. Phys. A Math. Theor. 46, 035202 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Bivariate second-order linear partial differential equations and orthogonal polynomial solutions. J. Math. Anal. Appl. 387, 1188–1208 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    I. Area, E. Godoy, J. Rodal, On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions. J. Math. Anal. Appl. 389, 165–178 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    I. Area, N.M. Atakishiyev, E. Godoy, J. Rodal, Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions. Appl. Math. Comput. 223, 520–536 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    N.M. Atakishiyev, M. Rahman, S.K. Suslov, On classical orthogonal polynomials. Constr. Approx. 11, 181–223 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    S. Bochner, Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29, 730–736 (1929)Google Scholar
  11. 11.
    C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 81 (Cambridge University Press, Cambridge, 2001)Google Scholar
  12. 12.
    G.K. Engelis, On some two–dimensional analogues of the classical orthogonal polynomials (in Russian). Latviı̆skiı̆ Matematic̆eskiı̆ Ez̆egodnik 15, 169–202 (1974)Google Scholar
  13. 13.
    C. Ferreira, J.L. López, P.J. Pagola, Asymptotic approximations between the Hahn-type polynomials and Hermite, Laguerre and Charlier polynomials. Acta Appl. Math. 103, 235–252 (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    M. Foupouagnigni, On difference equations for orthogonal polynomials on nonuniform lattices. J. Differ. Equ. Appl. 14, 127–174 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Foupouagnigni, M. Kenfack Nangho, S. Mboutngam, Characterization theorem for classical orthogonal polynomials on non-uniform lattices: the functional approach. Integral Transform. Spec. Funct. 22, 739–758 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Foupouagnigni, W. Koepf, M. Kenfack-Nangho, S. Mboutngam, On solutions of holonomic divided-difference equations on nonuniform lattices. Axioms 3, 404–434 (2014)zbMATHGoogle Scholar
  17. 17.
    J. Geronimo, P. Iliev, Bispectrality of multivariable Racah-Wilson polynomials. Constr. Approx. 31, 417–457 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    J.S. Geronimo, P. Iliev, Multivariable Askey-Wilson function and bispectrality. Ramanujan J. 24, 273–287 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    E. Godoy, A. Ronveaux, A. Zarzo, I. Area, On the limit relations between classical continuous and discrete orthogonal polynomials. J. Comput. Appl. Math. 91, 97–105 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    R.C. Griffiths, Orthogonal polynomials on the multinomial distribution. Austral. J. Stat. 13, 27–35 (1971). Corrigenda (1972) Austral. J. Stat. 14, 270Google Scholar
  21. 21.
    W. Hahn, Über Orthogonalpolynome, die q-Differenzengleichungen genügen. Math. Nachr. 2, 4–34 (1949)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Ch. Hermite, Extrait d’une lettre *) de M. Hermite à M. Borchardt. J. Crelle 64, 294–296 (1865). As indicated in the publication *)=“Une exposition plus détaillée du sujet traité dans cette lettre se trouve dans les Comptes Rendus de l’Académie des Sciences de Paris, année 1865, séances du 20 et 27 février, du 6 et 16 mars”. https://babel.hathitrust.org/cgi/pt?id=mdp.39015036985383
  23. 23.
    N.L. Johnson, S. Kotz, A.W. Kemp, Univariate Discrete Distributions (Wiley, New York, 1992)zbMATHGoogle Scholar
  24. 24.
    N.L. Johnson, S. Kotz, N. Balakrishnan, Discrete Multivariate Distributions (Wiley, New York, 1997)zbMATHGoogle Scholar
  25. 25.
    S. Karlin, J. McGregor, On some stochastic models in genetics, in Stochastic Models in Medicine and Biology, ed. by J. Gurland (University of Wisconsin Press, Madison, 1964), pp. 245–271Google Scholar
  26. 26.
    S. Karlin, J. McGregor, Linear growth models with many types and multidimensional Hahn polynomials, in Theory and Application of Special Functions. Proceedings of an Advanced Seminar, The University of Wisconsin, Madison (1975), pp. 261–288Google Scholar
  27. 27.
    R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Theirq-Analogues. Springer Monographs in Mathematics (Springer, Berlin, 2010)Google Scholar
  28. 28.
    T. Koornwinder, Two–variable analogues of the classical orthogonal polynomials, in Theory and Application of Special Functions, ed. by R. Askey. Proceedings of an Advanced Seminar, The University of Wisconsin–Madison (Academic, Cambridge, 1975), pp. 435–495Google Scholar
  29. 29.
    T.H. Koornwinder, The Askey scheme as a four-manifold with corners. Ramanujan J. 20, 409–439 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    M.A. Kowalski, The recursion formulas for orthogonal polynomials in n variables. SIAM J. Math. Anal. 13, 309–315 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    M.A. Kowalski, Orthogonality and recursion formulas for polynomials in n variables. SIAM J. Math. Anal. 13, 316–323 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    H.L. Krall, I.M. Sheffer, Orthogonal polynomials in two variables. Ann. Mat. Pura Appl. 4, 325–376 (1967)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    P. Lesky, Zweigliedrige Rekursionen für die Koeffizienten von Polynomlösungen Sturm-Liouvillescher q-Differenzengleichungen. Z. Angew. Math. Mech. 74, 497–500 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    A.S. Lyskova, Orthogonal polynomials in several variables. Sov. Math. Dokl. 43, 264–268 (1991)MathSciNetzbMATHGoogle Scholar
  35. 35.
    A.S. Lyskova, On some properties of orthogonal polynomials in several variables. Russ. Math. Surv. 52, 840–841 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    A.P. Magnus, Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, in Orthogonal Polynomials and Their Applications (Segovia, 1986), ed. by M. Alfaro et al., Lecture Notes in Mathematics, vol. 1329 (Springer, Berlin, 1988), pp. 261–278Google Scholar
  37. 37.
    A.P. Magnus, Special nonuniform lattice (SNUL) orthogonal polynomials on discrete dense sets of points. J. Comput. Appl. Math. 65, 253–265 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    F. Marcellán, A. Branquinho, J. Petronilho, Classical orthogonal polynomials: a functional approach. Acta Appl. Math. 34(3), 283–303 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    H. Miki, S. Post, L. Vinet, A. Zhedanov, A superintegrable finite oscillator in two dimensions with SU(2) symmetry. J. Phys. A: Math. Gen. 46, 1–13 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    G. Munschy, Résolution de l’équation de Schrödinger des atomes à deux électrons III. J. Phys. Radium 8, 552–558 (1957)CrossRefMathSciNetGoogle Scholar
  41. 41.
    G. Munschy, P. Pluvininage, Résolution de l’équation de Schrödinger des atomes à deux électrons II. J. Phys. Radium 8, 157–160 (1957)CrossRefGoogle Scholar
  42. 42.
    M. Njinkeu Sandjon, A. Branquinho, M. Foupouagnigni, I. Area, Characterizations of classical orthogonal polynomials on quadratic lattices. J. Differ. Equ. Appl. 23, 983–1002 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    P. Njionou Sadjang, W. Koepf, M. Foupouagnigni, On moments of classical orthogonal polynomials. J. Math. Anal. Appl. 424, 122–151 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, Berlin, 1991)CrossRefzbMATHGoogle Scholar
  45. 45.
    J. Proriol, Sur une famille de polynômes à deux variables orthogonaux dans un triangle. C.R. Acad. Sci. Paris 245, 2459–2461 (1957)Google Scholar
  46. 46.
    J. Rodal, I. Area, E. Godoy, Orthogonal polynomials of two discrete variables on the simplex. Integral Transform. Spec. Funct. 16, 263–280 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    J. Rodal, I. Area, E. Godoy, Linear partial difference equations of hypergeometric type: orthogonal polynomial solutions in two discrete variables. J. Comput. Appl. Math. 200, 722–748 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    J. Rodal, I. Area, E. Godoy, Structure relations for monic orthogonal polynomials in two discrete variables. J. Math. Anal. Appl. 340, 825–844 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  49. 49.
    A. Ronveaux, A. Zarzo, I. Area, E. Godoy, Transverse limits in the Askey tableau. J. Comput. Appl. Math. 99, 327–335 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    A. Ronveaux, A. Zarzo, I. Area, E. Godoy, Bernstein bases and Hahn-Eberlein orthogonal polynomials. Integral Transform. Spec. Funct. 7(1–2), 87–96 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    M.H. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series. Ellis Horwood Series: Mathematics and its Applications (Ellis Horwood, Chichester, 1985)Google Scholar
  52. 52.
    P.K. Suetin, Orthogonal Polynomials in Two Variables (Gordon and Breach Science Publishers, Amsterdam, 1999)zbMATHGoogle Scholar
  53. 53.
    D.D. Tcheutia, Y. Guemo Tefo, M. Foupouagnigni, E. Godoy, I. Area, Linear partial divided-difference equation satisfied by multivariate orthogonal polynomials on quadratic lattices. Math. Model. Nat. Pheno. 12, 14–43 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    D.D. Tcheutia, M. Foupouagnigni, Y. Guemo Tefo, I. Area, Divided-difference equation and three-term recurrence relations of some systems of bivariate q-orthogonal polynomials. J. Differ. Equ. Appl. 23, 2004–2036 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    M.V. Tratnik, Multivariable Meixner, Krawtchouk, and Meixner–Pollaczek polynomials. J. Math. Phys. 30, 2740–2749 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau—continuous families. J. Math. Phys. 32, 2065–2073 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  57. 57.
    M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau—discrete families. J. Math. Phys. 32, 2337–2342 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  58. 58.
    Y. Xu, Second order difference equations and discrete orthogonal polynomials of two variables. Intl. Math. Res. Not. 8, 449–475 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada II, E.E. Aeronáutica e do EspazoUniversidade de VigoOurenseSpain

Personalised recommendations