On an identity of Mahler (Q2473639)

The \(m\)-fold Euler-type integrals \[ F(z)=\idotsint_{[0,1]^m} \prod_{j=1}^m\frac{x_j^{a_j-1}(1-x_j)^{b_j-a_j-1}} {(1-zx_j\dotsb x_m)^{c_j}}\;dx_1\dotsb dx_m, \qquad | z|<1, \] where \(a_j\), \(b_j\) and \(c_j\) are integer parameters satisfying \(0<a_j<b_j\) (to ensure the convergence), specialized at the point \(z=1\), play an important role in the arithmetic study of the values of Riemann's zeta function \(\zeta(s)\) at integers \(s\geq2\); cf., for example [\textit{F. Beukers}, Bull. Lond. Math. Soc. 11, 268--272 (1979; Zbl 0421.10023); \textit{V. N.~Sorokin}, Sb. Math. 187, No.~12, 1819--1852 (1996); translation from Mat. Sb. 187, No. 12, 87--120 (1996; Zbl 0876.11035); \textit{G.~Rhin} and \textit{C.~Viola}, Acta Arith. 97, No.~3, 269--293 (2001; Zbl 1004.11042); \textit{W.~Zudilin}, J. Lond. Math. Soc. (2) 70, No.~1, 215--230 (2004; Zbl 1065.11054)]. A general theorem of \textit{S. A.~Zlobin} [Math. Notes 77, No.~5, 630--652 (2005); translation from Mat. Zametki 77, No. 5, 683--706 (2005; Zbl 1120.11030)] says that the integrals \(F(z)\) can be represented as linear forms, with polynomial coefficients, in the so-called generalized polylogarithms, and these are the forms used in the proofs of several results on the linear and algebraic independence of the values of polylogarithms, in particular, of~\(\zeta(s)\). The introductory part of the article reviews the corresponding cases. The very first ``Diophantine'' application of the integral \(F(z)\), namely, to estimate the irrationality measures of algebraic numbers and the transcendence measures of the logarithms of algebraic numbers, was given by \textit{K.~Mahler} in [Math. Ann. 105, No. 1, 267--276 (1931; Zbl 0002.18401); J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101); J. Reine Angew. Math. 166, 137--150 (1932; Zbl 0003.38805)]. A version of Mahler's original result gets a new proof in the article under review (Theorem~1), while Theorem~2 provides a generalization of Theorem~1 with certain sufficient conditions for the integers \(a_j,b_j,c_j\) to ensure an expansion of the form \[ z^{b_m-1}F(z) =\sum_{k=0}^dA_k(z)\frac{(-\log(1-z))^k}{k!}, \quad\text{where }A_k(z)\in\mathbb Q[z]. \]