Algebraic independence of modified reciprocal sums of products of Fibonacci numbers (Q875679)

In this paper the author establishes, using Mahler's method and Masser's vanishing theorem, the algebraic independence of reciprocal sums of products of Fibonacci numbers including slowly increasing factors in their numerators. Let \(\{F_n\}_{n\geq 0}\) be the sequence of Fibonacci numbers defined by (1) \(F_0 = 0\), \(F_1 = 1\), \(F_{n+2} = F_{n+1} + F_n\) \((n \geq 0)\). In this paper the author considers the arithmetic nature of the sums of series such as \[ \sum_{n=1}^\infty \frac{(-1)^n[\log_d n]}{F_nF_{n+k}}\qquad (d\in \mathbb N\setminus \{1\},\, k \in \mathbb N) \] and \[ \sum_{n=1}^\infty \frac{[\log_d n]}{F_nF_{n+2k}}\qquad (d\in \mathbb N\setminus \{1\},\, k \in \mathbb N), \] where \([x]\) denotes the largest integer not exceeding the real number \(x\). These sums are not only transcendental but also algebraically independent in contrast with the sums \(\sigma_k=\sum_{n=1}^\infty \frac{(-1)^n}{F_nF_{n+k}}\) (Brousseau) and \(\sigma_k^*=\sum_{n=1}^\infty \frac1{F_nF_{n+2k}}\) (Rabinowitz) which are algebraic numbers. Now let \(\{R_n\}_{n\geq 0}\) be the binary linear recurrence defined by \[ R_{n+2} = A_1R_{n+1} + A_2R_n\quad (n \geq 0),\tag{2} \] where \(A_1, A_2\) are nonzero integers with \(\Delta = A_1^2 + 4A_2 > 0\) and \(R_0, R_1\) are integers with \(R_0R_2 = R_1^2\) and \(A_1R_0(A_1R_0-2R_1) \leq 0\). We can express \(\{R_n\}_{n\geq 0}\) as follows: \(R_n = a\alpha^n + b\beta^n\) \((n\geq 0)\), where \(\alpha, \beta\) \((|\alpha| \geq |\beta|)\) are the roots of \(\Phi(X) = X^2-A_1X-A_2\) and \(a, b\in\mathbb Q(\sqrt{\Delta})\). It is easily seen that \(|\alpha| > |\beta| > 0\). Since \(R_0R_2 -R_1^2= ab\Delta\) and \(A_1R_0(A_1R_0 -2R_1) =(\alpha^2 -\beta^2)(b^2 -a^2)\), we see that \(|a| \geq |b| > 0\). Therefore \(\{R_n\}_{n\geq 0}\) is not a geometric progression and \(R_n \neq 0\) for any \(n\geq 1\). Then the author proves the following: Theorem 1. The numbers \[ \sum_{n=1}^\infty \frac{(-A_2)^n[\log_d n]}{R_nR_{n+k}}\qquad (d\in \mathbb N\setminus \{1\},\, k \in \mathbb N) \] are algebraically independent and so are the numbers \[ \sum_{n=1}^\infty \frac{A_2^n[\log_d n]}{R_nR_{n+2k}}\qquad (d\in \mathbb N\setminus \{1\},\, k \in \mathbb N). \] The above result especially holds when \(R_n\) equals the Fibonacci or Lucas numbers.