Skip to main content
SpringerLink
Log in

Adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In this article, we obtain the adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices, and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well. The particular solutions of the adjoint difference equation are then obtained. As an application of these particular solutions, we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices. In addition, we give another kind of fundamental theorems for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type, which are essentially new results and their expressions are different from the Suslov Theorem. Finally, we give an example to illustrate the application of the new fundamental theorems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical Orthogonal Polynomials of a Discrete Variable. Translated from the Russian. Springer Series in Computational Physics. Springer, Berlin (1991)

  2. Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics: A Unified Introduction with Applications. Translated from the Russian by Ralph P. Boas. Birkhauser Verlag, Basel (1988)

  3. Suslov, S.K.: On the theory of difference analogues of special functions of hypergeometric type. Russ. Math. Surv. 44, 227–278 (1989)

    Article  MathSciNet  Google Scholar 

  4. Wang, Z.X., Guo, D.R.: Special Functions. World Scientific Publishing, Singapore (1989)

    Book  Google Scholar 

  5. Andrews, G.E., Askey, R.: Classical orthogonal polynomials. In: Polynomes Orthogonaux et Applications, pp. 36–62. Springer, Berlin (1985)

  6. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)

  7. Askey, R., Wilson, J.A.: A set of orthogonal polynomials that generalize the Racah coefficients or 6j-symbols. SIAM J. Math. Anal. 10, 1008–1016 (1979)

    Article  MathSciNet  Google Scholar 

  8. Askey, R., Ismail, M.E.H.: Recurrence Relations, Continued Fractions and Orthogonal Polynomials. Mem. Am. Math. Soc. No. 300 (1984)

  9. Askey, R., Wilson, J.A.: Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials. Mem. Am. Math. Soc. No. 319 (1985)

  10. Ismail, M.E.H., Libis, C.A.: Contiguous relations, basic hypergeometric functions, and orthogonal polynomials I. J. Math. Anal. Appl. 141, 349–372 (1989)

    Article  MathSciNet  Google Scholar 

  11. Nikiforov, A.F., Urarov, V.B.: Classical orthogonal polynomials of a discrete variable on non-uniform lattices. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint, No. 17 (1983)

  12. Atakishiyev, N.M., Suslov, S.K.: Difference hypergeometric functions. In: Progress in Approximation Theory, pp. 1–35. Springer New York (1992)

  13. Atakishiyev, N.M., Suslov, S.K.: About one class of special function. Rev. Mex. Fis. 34(2), 152–167 (1988)

    MathSciNet  MATH  Google Scholar 

  14. George, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  15. Koornwinder, T.H.: q-Special functions, a tutorial. arXiv:math/9403216

  16. Foupouagnigni, M.: On difference equations for orthogonal polynomials on nonuniform lattices. J. Differ. Equ. Appl. 14, 127174 (2008)

    Article  MathSciNet  Google Scholar 

  17. Foupouagnigni, M., Koepf, W., Kenfack-Nangho, K., Mboutngam, S.: On solutions of holonomic divided-difference equations on nonuniform lattices. Axioms 2, 404434 (2013)

    Article  Google Scholar 

  18. Jia, L., Cheng, J., Feng, Z.: A q-analogue of Kummer’s equation. Electron J. Differ. Equ. 2017, 1–20 (2017)

    Article  MathSciNet  Google Scholar 

  19. Magnus, A.P.: Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points. In: Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994) 65, pp. 253–265 (1995)

  20. Swarttouw, R.F., Meijer, H.G.: A \(q\)-analogue of the Wronskian and a second solution of the Hahn-Exton \(q\)-Bessel difference equation. Proc. Am. Math. Soc. 120, 855–864 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Witte, N.S.: Semi-classical orthogonal polynomial systems on non-uniform lattices, deformations of the Askey table and analogs of isomonodromy. arXiv:1204.2328 (2011)

  22. Dreyfus, T.: \(q\)-Deformation of meromorphic solutions of linear differential equations. J. Differ. Equ. 259, 5734–5768 (2015)

    Article  MathSciNet  Google Scholar 

  23. Horner, J.M.: A note on the derivation of Rodrigues’ formulae. Am. Math. Monthly 70, 81–82 (1963)

    Article  MathSciNet  Google Scholar 

  24. Horner, J.M.: Generalizations of the formulas of Rodrigues and Schlafli. Am. Math. Monthly 71, 870–876 (1963)

    MathSciNet  MATH  Google Scholar 

  25. Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    Book  Google Scholar 

  26. Robin, W.: On the Rodrigues formula solution of the hypergeometric type differential equation. Int. Math. Forum 8, 1455–1466 (2013)

    Article  MathSciNet  Google Scholar 

  27. Bangerezako, G.: Variational calculus on \(q\)-nonuniform lattices. J. Math. Anal. Appl. 306, 161–179 (2005)

    Article  MathSciNet  Google Scholar 

  28. Álvarez-Nodarse, R., Cardoso, K.L.: On the properties of special functions on the liear-type lattices. J. Math. Anal. Appl. 405, 271–285 (2011)

    Article  Google Scholar 

  29. Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Hypergeometric type differential equations: second kind solutions and related integrals. J. Comput. Appl. Math. 157, 93–106 (2003)

    Article  MathSciNet  Google Scholar 

  30. Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Hypergeometric type \(q\)-difference equations: Rodrigues type representation for the second kind solution. J. Comput. Appl. Math. 173, 81–92 (2005)

    Article  MathSciNet  Google Scholar 

  31. Cheng, J.F., Jia, L.K.: Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices. arXiv:1811.07114v1 [math.CA] (2018)

  32. Rahman, M.: An integral representation of a \(_{10}\Phi _{9}\) and continuous bi-orthogonal \(_{10}\Phi _{9}\) rational functions. Can. J. Math. 38, 605–618 (1986)

    Article  Google Scholar 

  33. Koekoek, R., Lesky, P.E., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their \(q\)-analogues. Springer, Berlin (2010)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, People’s Republic of China

    Jinfa Cheng

  2. Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA, 71272, USA

    Weizhong Dai

Authors
  1. Jinfa Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weizhong Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinfa Cheng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported by the Fundamental Research Funds for the Central Universities of China, Grant Number 20720150006, and Natural Science Foundation of Fujian province of China, Grant Number 2016J01032.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, J., Dai, W. Adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices. Ramanujan J 53, 285–318 (2020). https://doi.org/10.1007/s11139-020-00298-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-020-00298-3

Keywords

Mathematics Subject Classification

Search

Navigation