On arithmetical properties of series (Q1280357)

Let \(q = p^{r_1}_1,\dots,p^{r_s}_s\), where \(p_1,\dots,p_s\) are prime numbers, and let \(\Omega_1,\dots,\Omega_s\) be the completions of algebraic closures of the fields \(\mathbb Q_{p_1},\dots,\mathbb Q_{p_s}\). It is assumed that the series \(f(x)\) represents a transcendental function in every field \(\Omega_1,\dots,\Omega_s\) and satisfies the functional equation \[ f(x) = P_0(x) + P_1(x)f(qx), \] where the degree of the polynomial \(P_1(x)\) is equal to \(\delta\). The author studies arithmetic properties of the values at algebraic points which are taken by the above mentioned series in some subcollection of the fields \(\Omega_j\), \(j = 1,2,\dots,t\). Theorem. Let \(d\in \mathbb N\) and \(\xi_1,\dots, \xi_m\) be algebraic integers such that \(|\xi_i|_{p_j}=1\). Suppose that the coefficients of \(P_0\) and \(P_1\) belong to an algebraic number field \(\mathbb K_0\). Denote \[ \ln q= S= \sum_{j=1}^s r_j\ln p_j, \qquad T= \sum_{j=1}^t r_j\ln p_j. \] Suppose that for each \(1,\dots, m\), the series \(f(\xi_i)\) converges in each field \(\Omega_1,\dots, \Omega_t\) to the same algebraic number \(\alpha_i\). Suppose also \([\mathbb K:\mathbb Q]=d\), where \(\mathbb K= \mathbb K_0 (\xi_1,\dots, \xi_m,\alpha_1,\dots, \alpha_m)\). Then \[ m\leq (4d(1+\delta) S/T+1)^2. \] Corollary. The statement of the theorem is valid for \(f(x)= \sum_{k=0}^\infty q^{k(k+1)/2} x^k\) with the estimate \(m\leq (8dS/T+1)\).