Skip to main content
SpringerLink
Log in

Sundry topics

  • Chapter
A Comprehensive Treatment of q-Calculus

  • 1446 Accesses

Abstract

This chapter contains applications of the previous one. Examples are two quadratic q-hypergeometric transformations. Most of the summation formulas are given both in q-hypergeometric form and in q-binomial coefficient form for convenience. We use the q-analogue of Euler’s mirror formula to prove a summation formula. We give another proof of the q-Dixon formula by means of yet another q-analogue of Kummer’s first summation formula. We find a q-analogue of Truesdell’s function and its functional equation. The Bailey transformation for q-series is of great theoretical interest. We find a q-Taylor formula with Lagrange remainder term. Finally, bilateral q-hypergeometric series are treated.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Agarwal, R.P.: Some relations between basic hypergeometric functions of two variables. Rend. Circ. Mat. Palermo (2) 3, 76–82 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, G.E.: On the q-analog of Kummer’s theorem and applications. Duke Math. J. 40, 525–528 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews, G.E.: On q-analogues of the Watson and Whipple summations. SIAM J. Math. Anal. 7(3), 332–336 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andrews, G.E., Askey, R.: A simple proof of Ramanujan’s summation of the 1 ψ 1. Aequ. Math. 18(3), 333–337 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Askey, R.: The q-gamma and q-beta functions. Appl. Anal. 8(2), 125–141 (1978/79)

    Article  MathSciNet  Google Scholar 

  7. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge (1935), reprinted by Stechert-Hafner, New York (1964)

    Google Scholar 

  8. Brown, B.M., Eastham, M.S.P.: A note on the Dixon formula for a finite hypergeometric series. J. Comput. Appl. Math. 194, 173–175 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlitz, L.: Some formulas of F.H. Jackson. Monatshefte Math. 73, 193–198 (1969)

    Article  MATH  Google Scholar 

  10. Chen, W.Y.C., Liu, Z.-G.: Parameter augmentation for basic hypergeometric series, I. In: Mathematical Essays in Honor of Gian-Carlo Rota, pp. 111–129. Birkhäuser, Basel (1998)

    Chapter  Google Scholar 

  11. Ernst, T.: A method for q-calculus. J. Nonlinear Math. Phys. 10(4), 487–525 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ernst, T.: Some results for q-functions of many variables. Rend. Semin. Mat. Univ. Padova 112, 199–235 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Ernst, T.: q-analogues of some operational formulas. Algebras Groups Geom. 23(4), 354–374 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Euler, L.: Introductio in Analysin Infinitorum, vol. 1. Lausanne (1748)

    Google Scholar 

  15. Exton, H.: Multiple Hypergeometric Functions and Applications. Mathematics and Its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [Wiley, Inc.], New York, 312 pp. (1976)

    Google Scholar 

  16. Exton, H.: Q-Hypergeometric Functions and Applications. Ellis Horwood, Chichester; Halsted Press [Wiley, Inc.], New York (1983)

    MATH  Google Scholar 

  17. Fine N. (Andrews, G.): Basic Hypergeometric Series and Applications. With a Foreword by George E. Andrews. Mathematical Surveys and Monographs, vol. 27. American Mathematical Society, Providence (1988)

    Google Scholar 

  18. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  20. Gauß, C.F.: Werke 2, 9–45 (1876).

    Google Scholar 

  21. Goldman, J., Rota, G.C.: The number of subspaces of a vector space. In: Recent Progress in Combinatorics, pp. 75–83. Academic Press, New York (1969)

    Google Scholar 

  22. Guo, V.: Elementary proofs of some q-identities of Jackson and Andrews-Jain. Discrete Math. 295(1–3), 63–74 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hahn, W.: Lineare geometrische Differenzengleichungen. Forschungszentrum Graz, Bericht 169 (1981)

    Google Scholar 

  24. Hardy, G.H.: Ramanujan. Cambridge University Press, Cambridge (1940). Reprinted by Chelsea, New York (1978)

    Google Scholar 

  25. Jackson, F.H.: On generalized functions of Legendre and Bessel. Trans. R. Soc. Edinb. 41, 1–28 (1904)

    Google Scholar 

  26. Jackson, F.H.: A q-form of Taylor’s theorem. Messenger 38, 62–64 (1909)

    Google Scholar 

  27. Jackson, F.H.: Certain q-identities. Q. J. Math. Oxf. Ser. 12, 167–172 (1941)

    Article  MATH  Google Scholar 

  28. Jing, S.-C., Fan, H.-Y.: q-Taylor’s formula with its q-remainder. Commun. Theor. Phys. 23(1), 117–120 (1995)

    MathSciNet  Google Scholar 

  29. Kaneko, M., Kurokawa, N., Wakayama, M.: A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math. 57(1), 175–192 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kim, Y., Rathie, A., Lee, C.: On q-Gauss’s second summation theorem. Far East J. Math. Sci. 17(3), 299–303 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Koekoek, J., Koekoek, R.: A note on the q-derivative operator. J. Math. Anal. Appl. 176(2), 627–634 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kummer, E.E.: Über die hypergeometrische Reihe …. J. Reine Angew. Math. 15, 39–83, 127–172 (1836)

    Article  MATH  Google Scholar 

  33. Petkovsek, M., Wilf, H., Zeilberger, D.: A=B. A.K. Peters, Wellesley (1996)

    Google Scholar 

  34. Rainville, E.D.: Special Functions. Chelsea Publishing Co., Bronx (1971). Reprint of 1960 first edition

    MATH  Google Scholar 

  35. Rajković, P.M., Stanković, M.S., Marinković, S.D.: Mean value theorems in q-calculus. Mat. Vestn. 54(3–4), 171–178 (2002). Proceedings of the 5th International Symposium on Mathematical Analysis and Its Applications, Niška Banja, 2002

    MATH  Google Scholar 

  36. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge (1966)

    Google Scholar 

  37. Srivastava, H.M.: Sums of a certain class of q-series. Proc. Jpn. Acad., Ser. A, Math. Sci. 65(1), 8–11 (1989)

    Article  MATH  Google Scholar 

  38. Srivastava, H.M., Karlsson, P.W.: Transformations of multiple q-series with quasi-arbitrary terms. J. Math. Anal. Appl. 231(1), 241–254 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Truesdell, C.: A Unified Theory of Special Functions Based Upon the Functional Equation \({\partial\over\partial Z} F(z,\alpha) = F(z,\alpha+1)\). Annals of Math. Studies, vol. 18. Princeton University Press, Princeton. London: Geoffrey Cumberlege. Oxford University Press (1948)

    MATH  Google Scholar 

  40. Verma, A.: A quadratic transformation of a basic hypergeometric series. SIAM J. Math. Anal. 11(3), 425–427 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vidunas, R.: Degenerate Gauss hypergeometric functions. Kyushu J. Math. 61(1), 109–135 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Watson, G.N.: A new proof of the Rogers-Ramanujan identities. J. Lond. Math. Soc. 4, 4–9 (1929)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Thomas Ernst

Authors
  1. Thomas Ernst
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel

About this chapter

Cite this chapter

Ernst, T. (2012). Sundry topics. In: A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0431-8_8

Download citation

Publish with us

Policies and ethics

Search

Navigation