Irrationality results for values of generalized Tschakaloff series. II. (Q1421304)

The main results of this paper are the following theorems. Theorem 1: Let \(q\) be an integer with \(| q|\geq 2\). Assume that \(\delta, \eta\) and \(c_0\) are constants with \(0<\delta<1\), \(0\leq\eta< \min(\tfrac 12,1-\delta)\) and \(0<c_0\). Suppose that \(\{s_n\}^\infty_{n=1}\), \(\{t_n\}^\infty_{n=1}\), \(\{u_n\}^\infty_{n= 1}\) and \(\{v_n\}^\infty_{n=1}\) are sequences of positive integers such that \(| s_n|\), \(| t_n|\) \(| v_n|\leq\exp(c_0n^\delta)\), \(| u_n|\leq\exp(c_0n^n)\) and \(\text{lcm}(t_0,t_1, \dots,t_n) \leq\exp (c_0n^{1+\eta})\). Then for every \(\alpha\in\mathbb{Q}\) the number \(f( \alpha)=\sum^\infty_{n=0} \frac {s_nu_0u_1,\dots, u_n}{t_nv_0 v_1, \dots,v_n}\frac {\alpha^n} {q^{n(n+1)/2}}\) is irrational. Theorem 2: Let \(q\) be an integer with \(| q|\geq 2\) and \(c_0\) be a positive constant Suppose that \(\{s_n\}^\infty_{n=1}\), \(\{t_n\}^\infty_{n= 1}\), \(\{u_n\}^\infty_{n=1}\) and \(\{v_n\}^\infty_{n=1}\) are sequences of nonzero integers such that \(| s_n|\), \(| t_n |\) \(|_n|\leq\exp (0(n))\), \(| u_n|\leq\exp(c_0)\) and \(lcm(t_0,t_1, \dots,t_n)\leq \exp(c_0n)\). Then for every \(\alpha\in\mathbb{Q}\) the number \(f(\alpha)=\sum^\infty_{n=0} \frac {s_nu_0u_1,\dots,u_n} {t_nv_0v_1,\dots, v_n}\frac{\alpha^n} {q^{n(n+1)/2}}\) is irrational. Theorem 3: Let \(m\) and \(q\) be integers with \(m\geq 2\) and \(| q | \geq 2\). Assume that \(\gamma\) and \(e_0\) are positive constants. Suppose that \(\{s_{n,j}\}^\infty_{n=1}\), \(\{t_{n,j}\}^\infty_{n=1}\) \((j=1,\dots,m)\), \(\{u_n\}^\infty_{n=2}\) and \(\{v_n\}^\infty_{n=1}\) are sequences of nonzero integers such that \(| s_{n,j}|\), \(| t_{n,j}|\) \(| v_n|\leq \exp(c_0n^\gamma)\), \(| u_n| \leq\exp(c_0)\) and \(\text{lcm}(t_0,t_1,\dots,t_n) \leq \exp(c_0n)\). Let \(\alpha_j\) \((j=1,\dots,m)\) be rational numbers such that \(|\alpha_i |/ |\alpha_j| \neq| q|^k\) for every \(i,j=1, |,m\) with \(i\neq j\) and any \(k\in\mathbb{Z}\). Then the numbers \(f(\alpha_j)= \sum^\infty_{n=0} \frac{s_{n,j} u_0u_1,\dots,u_n} {t_{n,j}v_0v_1, \dots,v_n} \frac{\alpha^n_j} {q^{n(n+1)/2}}\) \((j=1,\dots,m)\) are linearly independent over the rational numbers. Theorems 1 and 3 include also lower estimations. The proofs of these theorems are based on the Padé-type approximation. For Part I, see J. Number Theory 77, No. 1, 155--169 (1999; Zbl 0929.11020).