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Mathematical Notes

, Volume 79, Issue 1–2, pp 97–108 | Cite as

On an Identity of Mahler

  • Yu. V. Nesterenko
Article
  • 51 Downloads

Abstract

We prove that certain multiple integrals depending on the complex parameter z can be expressed as polynomials in z and ln(1 − z). Similar identities were first used by K. Mahler in connection with the proofs of certain results of the theory of transcendental numbers.

Key words

Mahler's identity multiple integral transcendental numbers generalized polylogarithm Pade approximation Apery approximation Diophantine approximation 

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References

  1. 1.
    V. V. Zudilin, “Algebraic relations for multiple zeta values,” Uspekhi Mat. Nauk [Russian Math. Surveys], 58 (2003), no. 1, 3–32.zbMATHMathSciNetGoogle Scholar
  2. 2.
    E. A. Ulanskii, “Identities for generalized polylogarithms,” Mat. Zametki [Math. Notes], 73 (2003), no. 4, 613–624.zbMATHMathSciNetGoogle Scholar
  3. 3.
    V. N. Sorokin, “On Apery's theorem,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1998), no. 3, 48–53.Google Scholar
  4. 4.
    V. N. Sorokin, “On the degree of transcendency of the number π2,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 187 (1996), no. 12, 87–120.zbMATHMathSciNetGoogle Scholar
  5. 5.
    Yu. V. Nesterenko, “Integral identities and constructions for joint approximations to the values of of the Riemann zeta function,” in: Proceedings of the IV International Conference “Problems of Current Interest in Number Theory and Its Applications” (Tula, 2001) [in Russian], Izd. Moskov. Univ., Moscow, 2002, pp. 115–132.Google Scholar
  6. 6.
    T. Rivoal, “La fonction zeta de Riemann prend une infinite de valeurs irrationelles aux entiers impairs,” C. R. Acad. Sci. Paris Ser. 1 Math., 331 (2000), no. 4, 267–270.zbMATHMathSciNetGoogle Scholar
  7. 7.
    S. A. Zlobin, “Expansions of multiple integrals in linear forms,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 398 (2004), no. 5, 595–598.MathSciNetGoogle Scholar
  8. 8.
    S. A. Zlobin, “Integrals expressible as linear forms in generalized polylogarithms” Mat. Zametki [Math. Notes], 71 (2002), no. 5, 782–787.zbMATHMathSciNetGoogle Scholar
  9. 9.
    K. Mahler, “Ein Beweis des Thue—Siegelschen Satzes uber die Approximation algebraischer Zahlen fur binomische Gleichungen,” Math. Ann., 105 (1931), 267–276.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    A. Baker, “Simultaneous rational approximations to certain algebraic numbers,” Proc. Cambridge Philos. Soc., 63 (1967), 693–702.zbMATHMathSciNetGoogle Scholar
  11. 11.
    K. Mahler, “Zur Approximation der Exponentialfunction und des Logarithmus,” J. Reine Angew. Math., 166 (1932), 118–150.Google Scholar
  12. 12.
    Y. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, 1975; Russian translation: Mir, Moscow, 1980.Google Scholar
  13. 13.
    H. Jager, “A multidimentional generalization of the Pade table,” Indag. Math., 26 (1964), 193–249.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Yu. V. Nesterenko
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

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