Algebraic independence of reciprocal sums of powers of certain Fibonacci-type numbers

Abstract

The Fibonacci-type numbers in the title look like $R_n=g_1\gamma_1^n+ g_2\gamma_2^n$ and $S_n=h_1\gamma_1^n+h_2\gamma_2^n$ for any $n\in\mathbb{Z}$, where the $g$'s, $h$'s, and $\gamma$'s are given algebraic numbers satisfying certain natural conditions. For fixed $k\in\mathbb{Z}_{>0}$, and for fixed non-zero periodic sequences $(a_h),(b_h),(c_h)$ of algebraic numbers, the algebraic independence of the series \[ \sum_{h=0}^\infty \frac{a_h}{\gamma_1^{kr^h}}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{b_h}{(R_{kr^h+\ell})^m}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{c_h}{(S_{kr^h+\ell})^m} \qquad \big((\ell,m,r)\in\mathbb{Z}\times \mathbb{Z}_{>0}\times\mathbb{Z}_{>1}\big) \] is studied. Here the main tool is Mahler's method which reduces the investigation of the algebraic independence of numbers (over $\mathbb{Q}$) to that of functions (over the rational function field) if they satisfy certain types of functional equations.