Skip to main content
SpringerLink
Log in

Partial q-analogues of the Pfaff/Cauchy derivative formula

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

The Pfaff/Cauchy derivative identities are generalizations of Leibniz formula for the nth derivative of a product of two functions. In this paper, we first derive three generalized forms of the q-Leibniz formula. The results are also partial q-analogues of the Pfaff/Cauchy derivative formulae. Then we give some applications and several q-identities are obtained.

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. Andrews, G.E.: Identities in combinatorics. II: A q-analog of the Lagrange inversion theorem. Proc. Am. Math. Soc. 53, 240–245 (1975)

    MATH  Google Scholar 

  2. Carlitz, L.: Some q-expansion theorems. Glas. Mat. 8, 205–214 (1973)

    Google Scholar 

  3. Cauchy, A.L.: Application du calcul des résidus a la sommation de plusieurs suites. In: Exercices de Mathématiques, vol. 1 (1826). Œuvres complès d’Augustin Cauchy, 2nd series, vol. 6, pp. 62–73. Gauthier-Villars, Paris (1887)

    Google Scholar 

  4. Chu, W.C.: Leibniz inverse series relations and Pfaff–Cauchy derivative identities. Rend. Mat. 29, 209–221 (2008)

    Google Scholar 

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

    MATH  Google Scholar 

  6. Gould, H.W.: Two remarkable extensions of the Leibniz differentiation formula. Bul. Inst. Politeh. Iaşi (N.S.) 18, 47–53 (1972)

    Google Scholar 

  7. Jacobi, C.G.J.: Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math. 56, 149–165 (1859)

    Article  MATH  Google Scholar 

  8. Johnson, W.P.: The Pfaff/Cauchy derivative identities and Hurwitz type extensions. Ramanujan J. 13, 167–201 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lagrange, J.L.: Nouvelle méthode pour résoudre les équations littérales par le moyen des séries. Mém. Acad. R. Sci. B.-Lett. Berl. Année 24, 251–326 (1770). Œuvres de Lagrange, vol. 3, pp. 5–73. Gauthier-Villars, Paris (1869)

    Google Scholar 

  10. Krattenthaler, C.: A new q-Lagrange formula and some applications. Proc. Am. Math. Soc. 90, 338–344 (1984)

    MathSciNet  MATH  Google Scholar 

  11. Pfaff, J.F.: Allgemeine Summation einer Reihe, worinn hohere differenziale vorkommen. In: Archiv der Reinen und Angewandten Mathematik, vol. 1, pp. 337–347 (1795)

    Google Scholar 

  12. Ruscheweyh, S.: Eine Verallgemeinerung der Leibnizschen Produktregel. Math. Nachr. 58, 241–245 (1973)

    Article  MathSciNet  Google Scholar 

  13. Walker, J.J.: Theorems in the calculus of operations. Proc. Lond. Math. Soc. 11, 108–113 (1880)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, P.R. China

    Peipei Tang & De-Yin Zheng

Authors
  1. Peipei Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. De-Yin Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to De-Yin Zheng.

Additional information

This work was partially supported by the Natural Science Foundation of Zhejiang Province of China (No. Y7080320), Scientific Research Foundation of Hangzhou normal university (No. HSKQ0041).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tang, P., Zheng, DY. Partial q-analogues of the Pfaff/Cauchy derivative formula. Ramanujan J 28, 239–252 (2012). https://doi.org/10.1007/s11139-011-9366-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-011-9366-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation