Mathematical Notes

, Volume 77, Issue 5–6, pp 630–652 | Cite as

Expansion of Multiple Integrals in Linear Forms

  • S. A. Zlobin
Article

Abstract

We prove general theorems on expansions of multiple integrals in linear forms in generalized polylogarithms with coefficients that are rational functions.

Key words

linear form multiple integral Riemann zeta function generalized polylogarithm Beukers integral rational function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    F. Beukers, “A note on the irrationality of ζ(2) and ζ(3),” Bull. London Math. Soc., 11 (1979), no. 3, 268–272.Google Scholar
  2. 2.
    O. N. Vasilenko, “Formulas for the values of of the Riemann zeta function at integer points,” in: Abstracts of the Conference “Number Theory and Its Applications” (Tashkent, September 26–28, 1990) [in Russian], Tashkent State Pedagogical Institute, 1990, p. 27.Google Scholar
  3. 3.
    D. V. Vasilyev (Vasil’ev), On Small Linear Forms for the Values of the Riemann Zeta Function at Odd Points, Preprint no. 1 (558), Nat. Acad. Sci. Belarus, Institute Math., Minsk, 2001.Google Scholar
  4. 4.
    V. V. Zudilin, “Perfectly balanced hypergeometric series and multiple integrals,” Uspekhi Mat. Nauk [Russian Math. Surveys], 57 (2002), no. 4, 177–178.Google Scholar
  5. 5.
    V. N. Sorokin, “The Apery theorem,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1998), no. 3, 48–52.Google Scholar
  6. 6.
    V. N. Sorokin, “On the measure of transcendence of the number π2,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 187 (1996), no. 12, 87–120.Google Scholar
  7. 7.
    S. A. Zlobin, “Integrals expressible as linear forms in generalized polylogarithms,” Mat. Zametki [Math. Notes], 71 (2002), no. 5, 782–787.Google Scholar
  8. 8.
    S. Fischler, “Formes lineaires en polyzetas et integrales multiples,” C. R. Acad. Sci. Paris Ser. I Math., 335 (2002), 1–4.Google Scholar
  9. 9.
    S. A. Zlobin, “On certain integral identities,” Uspekhi Mat. Nauk [Russian Math. Surveys], 57 (2002), no. 3, 153–154.Google Scholar
  10. 10.
    Hoang Ngoc Minh and M. Petitot, and J. van der Hoeven, “Shuffle algebra and polylogarithms,” Discrete Math., 225 (2000), no. 1–3, 217–230.CrossRefGoogle Scholar
  11. 11.
    E. A. Ulanskii, “Identities for generalized polylogarithms,” Mat. Zametki [Math. Notes], 73 (2003), no. 4, 613–624.Google Scholar
  12. 12.
    A. N. Kolmogorov and S. V. Fomin, Elements of Function Theory and Functional Analysis [in Russian], Nauka, Moscow, 1989.Google Scholar
  13. 13.
    D. V. Vasil’ev, “Some formulas for the Riemann zeta function at integer points,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1996), no. 1, 81–84.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. A. Zlobin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations