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


SIGMA 8 (2012), 042, 30 pages      arXiv:1107.2423      https://doi.org/10.3842/SIGMA.2012.042

On the Orthogonality of q-Classical Polynomials of the Hahn Class

Renato Álvarez-Nodarse a, Rezan Sevinik Adıgüzel b and Hasan Taşeli b
a) IMUS & Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla, Spain
b) Department of Mathematics, Middle East Technical University (METU), 06531, Ankara, Turkey

Received July 29, 2011, in final form July 02, 2012; Published online July 11, 2012

Abstract
The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation on a q-linear lattice by means of a qualitative analysis of the q-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every possible rational form of the polynomial coefficients in the q-Pearson equation, together with various relative positions of their zeros, to describe a desired q-weight function supported on a suitable set of points. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known q-polynomials of the Hahn class to a larger set of their parameters.

Key words: q-polynomials; orthogonal polynomials on q-linear lattices; q-Hahn class.

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References

  1. Álvarez-Nodarse R., On characterizations of classical polynomials, J. Comput. Appl. Math. 196 (2006), 320-337.
  2. Álvarez-Nodarse R., Polinomios hipergeométricos clásicos y q-polinomios, Monographs of the "García de Galdeano" Mathematics Seminar, Vol. 26, Universidad de Zaragoza Seminario Matematico "Garcia de Galdeano", Zaragoza, 2003.
  3. Álvarez-Nodarse R., Arvesú J., On the q-polynomials in the exponential lattice x(s)=c1qs+c3, Integral Transform. Spec. Funct. 8 (1999), 299-324.
  4. Álvarez-Nodarse R., Atakishiyev N.M., Costas-Santos R.S., Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra, J. Phys. A: Math. Gen. 38 (2005), 153-174, arXiv:1003.4853.
  5. Álvarez-Nodarse R., Medem J.C., q-classical polynomials and the q-Askey and Nikiforov-Uvarov tableaus, J. Comput. Appl. Math. 135 (2001), 197-223.
  6. Álvarez-Nodarse R., Sevinik Adgüzel R., Ta seli H., The orthogonality of q-classical polynomials of the Hahn class: a geometrical approach, arXiv:1107.2423.
  7. Andrews G.E., Askey R., Classical orthogonal polynomials, in Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Lecture Notes in Math., Vol. 1171, Springer, Berlin, 1985, 36-62.
  8. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  9. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319.
  10. Atakishiyev N.M., Klimyk A.U., Wolf K.B., A discrete quantum model of the harmonic oscillator, J. Phys. A: Math. Theor. 41 (2008), 085201, 14 pages, arXiv:0711.3089.
  11. Atakishiyev N.M., Rahman M., Suslov S.K., On classical orthogonal polynomials, Constr. Approx. 11 (1995), 181-226.
  12. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York, 1978.
  13. Dehesa J.S., Nikiforov A.F., The orthogonality properties of q-polynomials, Integral Transform. Spec. Funct. 4 (1996), 343-354.
  14. Elaydi S., An introduction to difference equations, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2005.
  15. Fine N.J., Basic hypergeometric series and applications, Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI, 1988.
  16. García A.G., Marcellán F., Salto L., A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math. 57 (1995), 147-162.
  17. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
  18. Grünbaum F.A., Discrete models of the harmonic oscillator and a discrete analogue of Gauss' hypergeometric equation, Ramanujan J. 5 (2001), 263-270.
  19. Hahn W., Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nachr. 2 (1949), 4-34.
  20. Kac V., Cheung P., Quantum calculus, Universitext, Springer-Verlag, New York, 2002.
  21. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  22. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998, http://aw.twi.tudelft.nl/~koekoek/askey/.
  23. Koornwinder T.H., Compact quantum groups and q-special functions, in Representations of Lie Groups and Quantum Groups (Trento, 1993), Pitman Res. Notes Math. Ser., Vol. 311, Longman Sci. Tech., Harlow, 1994, 46-128.
  24. Koornwinder T.H., Orthogonal polynomials in connection with quantum groups, in Orthogonal polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, 257-292.
  25. Marcellán F., Medem J.C., q-classical orthogonal polynomials: a very classical approach, Electron. Trans. Numer. Anal. 9 (1999), 112-127.
  26. Marcellán F., Petronilho J., On the solution of some distributional differential equations: existence and characterizations of the classical moment functionals, Integral Transform. Spec. Funct. 2 (1994), 185-218.
  27. Medem J.C., Álvarez-Nodarse R., Marcellán F., On the q-polynomials: a distributional study, J. Comput. Appl. Math. 135 (2001), 157-196.
  28. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
  29. Nikiforov A.F., Uvarov V.B., Classical orthogonal polynomials of a discrete variable on nonuniform nets, Inst. Prikl. Mat. M.V. Keldysha Akad. Nauk SSSR, Moscow, 1983, Preprint no. 17, 34 pages (in Russian).
  30. Nikiforov A.F., Uvarov V.B., Polynomial solutions of hypergeometric type difference equations and their classification, Integral Transform. Spec. Funct. 1 (1993), 223-249.
  31. Nikiforov A.F., Uvarov V.B., Special functions of mathematical physics: a unified introduction with applications, Birkhäuser Verlag, Basel, 1988.
  32. Suslov S.K., On the theory of difference analogues of special functions of hypergeometric type, Russ. Math. Surv. 44 (1989), 227-278.
  33. Vilenkin N.J., Klimyk A.U., Representation of Lie groups and special functions. Vol. 3. Classical and quantum groups and special functions, Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht, 1992.


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