Advertisement

The Ramanujan Journal

, Volume 48, Issue 1, pp 63–73 | Cite as

A note on q-difference equations for Ramanujan’s integrals

  • Jian CaoEmail author
Article
  • 115 Downloads

Abstract

This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to obtain the expression of certain q-orthogonal polynomials as solutions of q-difference equations. In addition, we show how to generalize Ramanujan’s integrals by the technique of q-difference equation. More over, we find two generalized q-Chu–Vandermonde formulas from the perspective of the method of q-difference equations.

Keywords

Solutions of q-difference equation Generating function Al-Salam–Carlitz polynomial Ramanujan’s integral 

Mathematics Subject Classification

05A30 11B65 33D15 33D45 39A13 

Notes

Acknowledgements

The author would like to thank the referees and editors for their many valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11501155).

References

  1. 1.
    Al-Salam, W.A., Carlitz, L.: Some orthogonal \(q\) -polynomials. Math. Nachr. 30, 47–61 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, G.E.: Applications of basic hypergeometric series. SIAM Rev. 16, 441–484 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Askey, R.: Two integrals of Ramanujan. Proc. Am. Math. Soc. 85, 192–194 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Askey, R., Ismail, M.E.H.: The very well-poised \({}_6\psi _6\). Proc. Am. Math. Soc. 77, 218–222 (1979)zbMATHGoogle Scholar
  5. 5.
    Atakishiyev, N.M., Feinsilver, P.: Two Ramanujan’s integrals with a complex parameter. In: Atakishiyev, N.M., Seligman, T.H., Wolf, K.B. (eds.), Proceedings of the IV Wigner Symposium, Guadalajara, Mexico, August 7–11, 1995, pp. 406–412. World Scientific, Singapore (1996)Google Scholar
  6. 6.
    Cao, J.: A note on \(q\) -integrals and certain generating functions. Stud. Appl. Math. 131, 105–118 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cao, J.: A note on generalized \(q\) -difference equations for \(q\) -beta and Andrews-Askey integral. J. Math. Anal. Appl. 412, 841–851 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carlitz, L.: Generating functions for certain \(q\) -orthogonal polynomials. Collectanea Math. 23, 91–104 (1972)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. 2nd edn, Encyclopedia Mathematics and its Applications vol. 96, Cambridge University Press, Cambridge (2004)Google Scholar
  10. 10.
    Gunning, R.: Introduction to Holomorphic Functions of Several Variables. In: Function theory, vol. 1, Wadsworth and Brooks/Cole, Belmont (1990)Google Scholar
  11. 11.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, paperback edn. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  12. 12.
    Koekoek, R., Swarttouw, R.F.: The Askey scheme of hypergeometric orthogonal polynomials and its \(q\) -analogue, Technical Report, pp. 98–17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft (1998)Google Scholar
  13. 13.
    Koekoek, R., lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and their \(q\) -Analogues, Springer Monographs in Mathematics, Springer, Berlin (2010)Google Scholar
  14. 14.
    Liu, Z.-G.: Two \(q\)- difference equations and \(q\) -operator identities. J. Differ. Equ. Appl. 16, 1293–1307 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, Z.-G.: An extension of the non-terminating \({}_6\phi _5\) summation and the Askey–Wilson polynomials. J. Differ. Equ. Appl. 17, 1401–1411 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Liu, Z.-G.: On the q-partial Differential Equations and q-series. In: The legacy of Srinivasa Ramanujan, vol. 20, pp. 213–250, Ramanujan Mathematical Society Lecture Note Series, Mysore (2013)Google Scholar
  17. 17.
    Liu, Z.-G., Zeng, J.: Two expansion formulas involving the Rogers-Szegö polynomials with applications. Int. J. Number Theory 11, 507–525 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, Z.-G.: A \(q\) -extension of a partial differential equation and the Hahn polynomials. Ramanujan J. 38, 481–501 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Malgrange, B.: Lectures on the Theory of Functions of Several Complex Variables. Springer, Berlin (1984)zbMATHGoogle Scholar
  20. 20.
    Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, American Mathematical Society Colloquium Publications, vol. 54, (part 1), AMS, Providence (2005)Google Scholar
  21. 21.
    Szegö, G.: Beitrag zur theorie der thetafunktionen. Sitz Preuss. Akad. Wiss. Phys. Math. Ki. 19, 242–252 (1926)zbMATHGoogle Scholar
  22. 22.
    Taylor, J.: Several complex variables with connections to algebraic geometry and lie groups, Graduate Studies in Mathematics, American Mathematical Society, Providence, vol. 46 (2002)Google Scholar
  23. 23.
    Wilf, H.S.: generatingfunctionology. Academic Press, San Diego (1994)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouPeople’s Republic of China

Personalised recommendations