Linear independence of values of functions satisfying Mahler functional equations (Q1276003)

The main result of the paper is as follows. Let a system of functions \(f_1(z),\dots,f_m(z)\) be given in the form of a Taylor power series in \(z\) with rational coefficients and satisfy the conditions: (a) the functions \(f_1(z),\dots,f_m(z)\) together with 1 are linearly independent over the field \(\mathbb C(z)\); (b) the functional equation \(\overline f(z) = A(z)\overline f(z^\rho) + \overline B(z)\), \(\rho\in \mathbb Z\), \(\rho\geq 2\) holds, where \(\overline f(z) = (f_1(z),\dots,f_m(z))^T\), \(A(z)\) and \(\overline B(z)\) are an \(m\times m\)-matrix and an \(m\)-vector consisting of the rational functions; (c) there exists a rational number \(\alpha = a/b\), \(0<|a|^{m+1}<|b|\), such that for \(z=\alpha\) all elements of the matrices \(A(z^{\rho^k})\) and vectors \(\overline B(z^{\rho^k})\) are defined and the determinants of the matrices do not vanish. Then the numbers \(1, f_1(z),\dots,f_m(z)\) are linearly independent over \(\mathbb Q\).