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

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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.

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References

  1. Al-Salam, W.A., Carlitz, L.: Some orthogonal \(q\) -polynomials. Math. Nachr. 30, 47–61 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, G.E.: Applications of basic hypergeometric series. SIAM Rev. 16, 441–484 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Askey, R.: Two integrals of Ramanujan. Proc. Am. Math. Soc. 85, 192–194 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Askey, R., Ismail, M.E.H.: The very well-poised \({}_6\psi _6\). Proc. Am. Math. Soc. 77, 218–222 (1979)

    MATH  Google Scholar 

  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)

  6. Cao, J.: A note on \(q\) -integrals and certain generating functions. Stud. Appl. Math. 131, 105–118 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlitz, L.: Generating functions for certain \(q\) -orthogonal polynomials. Collectanea Math. 23, 91–104 (1972)

    MathSciNet  MATH  Google Scholar 

  9. Gasper, G., Rahman, M.: Basic Hypergeometric Series. 2nd edn, Encyclopedia Mathematics and its Applications vol. 96, Cambridge University Press, Cambridge (2004)

  10. Gunning, R.: Introduction to Holomorphic Functions of Several Variables. In: Function theory, vol. 1, Wadsworth and Brooks/Cole, Belmont (1990)

  11. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, paperback edn. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  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)

  13. Koekoek, R., lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and their \(q\) -Analogues, Springer Monographs in Mathematics, Springer, Berlin (2010)

  14. Liu, Z.-G.: Two \(q\)- difference equations and \(q\) -operator identities. J. Differ. Equ. Appl. 16, 1293–1307 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  17. Liu, Z.-G., Zeng, J.: Two expansion formulas involving the Rogers-Szegö polynomials with applications. Int. J. Number Theory 11, 507–525 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, Z.-G.: A \(q\) -extension of a partial differential equation and the Hahn polynomials. Ramanujan J. 38, 481–501 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Malgrange, B.: Lectures on the Theory of Functions of Several Complex Variables. Springer, Berlin (1984)

    MATH  Google Scholar 

  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)

  21. Szegö, G.: Beitrag zur theorie der thetafunktionen. Sitz Preuss. Akad. Wiss. Phys. Math. Ki. 19, 242–252 (1926)

    MATH  Google Scholar 

  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)

  23. Wilf, H.S.: generatingfunctionology. Academic Press, San Diego (1994)

    MATH  Google Scholar 

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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).

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  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, Zhejiang, People’s Republic of China

    Jian Cao

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  1. Jian Cao
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Correspondence to Jian Cao.

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Dedicated to David Goss.

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Cao, J. A note on q-difference equations for Ramanujan’s integrals. Ramanujan J 48, 63–73 (2019). https://doi.org/10.1007/s11139-017-9987-1

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  • DOI: https://doi.org/10.1007/s11139-017-9987-1

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