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Expansion of multiple integrals in linear forms. (English) Zbl 1120.11030
Math. Notes 77, No. 5, 630-652 (2005); translation from Mat. Zametki 77, No. 5, 683-706 (2005).
After Apéry’s proof of the irrationality of \(\zeta(3)\), F. Beukers in 1979 gave an alternative argument involving the integrals \[ \int_0^1\int_0^1\int_0^1\frac{x^n (1-x)^ny^n (1-y)^n z^n(1-z)^n}{ \bigl(1-z(1-xy)\bigr)^{n+1}}\, dx\, dy\, dz \] [F. Beukers, Bull. Lond. Math. Soc. 11, 268–272 (1979; Zbl 0421.10023)]. Various generalizations have been introduced, especially by O. N. Vasilenko, D. V. Vasiliev, V. V. Zudilin and V. N. Sorokin.
In the paper under review the author considers integrals of the form \[ \int_{[0,1]^m} \frac{ \prod_{i=1}^m x_i^{a_i-1} (1-x_i)^{b_i-a_i-1} }{\prod_{j=1}^\ell \bigl(1-zx_1x_2\cdots x_{r_j}\bigr)^{c_j}} \,dx_1\cdots dx_m. \] Under suitable assumptions on the parameters \(a_1,b_1,\ldots,a_m,b_m\), \(c_1,r_1,\ldots,c_\ell,r_\ell\), he shows that such an integral is a linear combination of values of generalized polylogarithms.
Further related results on these integrals are due to J. Cresson, S. Fischler and T. Rivoal [Séries hypergéométriques multiples et polyzêtas, MathArXiv. preprint math.NT/0609743] and to S. Fischler [Multiple series connected to Hoffman’s conjecture on multiple zeta values, MathArXiv. preprint math.NT/0609799].

11J82 Measures of irrationality and of transcendence
11G55 Polylogarithms and relations with \(K\)-theory
33B30 Higher logarithm functions
Full Text: DOI
[1] F. Beukers, ”A note on the irrationality of \(\zeta\)(2) and \(\zeta\)(3),” Bull. London Math. Soc., 11 (1979), no. 3, 268–272. · Zbl 0421.10023 · doi:10.1112/blms/11.3.268
[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.
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[8] S. Fischler, ”Formes lineaires en polyzetas et integrales multiples,” C. R. Acad. Sci. Paris Ser. I Math., 335 (2002), 1–4.
[9] S. A. Zlobin, ”On certain integral identities,” Uspekhi Mat. Nauk [Russian Math. Surveys], 57 (2002), no. 3, 153–154. · Zbl 1058.11057
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[11] E. A. Ulanskii, ”Identities for generalized polylogarithms,” Mat. Zametki [Math. Notes], 73 (2003), no. 4, 613–624.
[12] A. N. Kolmogorov and S. V. Fomin, Elements of Function Theory and Functional Analysis [in Russian], Nauka, Moscow, 1989. · Zbl 0672.46001
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