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