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.
Citation
Peter Bundschuh. Keijo Väänänen. "Algebraic independence of reciprocal sums of powers of certain Fibonacci-type numbers." Funct. Approx. Comment. Math. 53 (1) 47 - 68, September 2015. https://doi.org/10.7169/facm/2015.53.1.4
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