Algebraic independence of Fibonacci reciprocal sums associated with Newton's method (Q1881437)

The main results of this paper are the following theorems. Theorem 1: Let \(A_1\) and \(A_2\) are two integers such that \(A_1> 0\), \(A_2\neq 0\) and \(A_2+ 4A_2> 0\). Let \(\{U_n\}^\infty_{n=0}\) be the binary linear recurrence such that \(U_0= 1\), \(U_1= 1\) and \(U_{n+2}= A_1 U_{n+1}+ A_2U_n\) for all \(n\geq 0\). Then the numbers \(\sum^\infty_{n=2} {(-A_2)^n[\log_2 n]\over U_{n+d} U_{n+d+1}}\), where \(d\) are nonnegative integers, are algebraically independent. Theorem 2: Let \(A_1\) and \(A_2\) are two integers such that \(A_1,A_2\neq 0\) and \(A_2+ 4A_2> 0\). Let \(\{U_n\}^\infty_{n=0}\) be the binary linear recurrence such that \(U_0\), \(U_1\) are integers, \(U_0 U_2\neq U^2_1\), \(A_1 U_0(A_1 U_0- 2U_1)\leq 0\) and \(U_{n+2}= A_1 U_{n+1}+ A_2 U_n\) for all \(n\geq 0\). Then the numbers \(\sum^\infty_{n=2} {(-A_2)^n[\log_s n]\over U_{n+d} U_{n+d+1}}\), where \(s\geq 2\) and \(d\geq 0\) are integers, are algebraically independent. A criterion for special series to be transcendental is included. The proofs are based on Newton's method.