Algebraic independence of reciprocal sums of powers of certain Fibonacci-type numbers (Q1744918)

Let \(\gamma_1\) and \(\gamma_2\) be non-zero algebraic numbers satisfying \(|\gamma_1|>\max(1,|\gamma_2|)\), and define \(R_n=g_1\gamma_1^n+g_2\gamma_2^n\) and \(S_n=h_1\gamma_1^n+h_2\gamma_2^n\) for some non-zero algebraic numbers \(g_1\), \(g_2\), \(h_1\) and \(h_2\), so that the special case is chosen as Fibonacci numbers \((R_n)=(F_n)\) and Lucas numbers \((S_n)=(L_n)\) . For integers \(k\), \(\ell\), \(m\) and \(r\) with \(k,m\geq 1\) and \(r\geq 2\) and for fixed non-zero periodic sequences \((a_h)\), \((b_h)\) and \((c_h)\) of algebraic numbers, consider the series of the forms \[ \sum_{h=0}^\infty\frac{a_h}{\gamma_1^{k r^h}},\quad \sum_{h=0}^\infty{}'\frac{b_h}{(R_{k r^h+l\ell})^m},\quad \sum_{h=0}^\infty{}'\frac{c_h}{(S_{k r^h+\ell})^m}. \] Here, \(\sum_{h\geq 0}'\) is taken over the non-negative integers \(h\) such that \(k r^h+\ell\geq 0\) and \(R_{k r^h+\ell}\neq 0\) (or \(S_{k r^h+\ell}\neq 0\)). In this paper by applying the ideas and results in [the authors, J. Aust. Math. Soc. 98, No. 3, 289--310 (2015; Zbl 1356.11044); \textit{K. Nishioka}, Monatsh. Math. 136, No. 2, 123--141 (2002; Zbl 1020.11050); \textit{K. Nishioka} et al., Math. Nachr. 202, 97--108 (1999; Zbl 0939.11028)], linear independence and then algebraic independence of these Mahler-type functions are obtained. The main tool is Mahler's method reducing the investigation of the algebraic independence of the numbers over \(\mathbb Q\) to that of functions over the rational function field.