Duplications In the \(k\)-generalized Fibonacci sequences (Q2045887)

Let \(k \geq 2\) be an integer. The \(k\)-generalized Fibonacci sequence \(({F_n}^{(k)})_{n \in\mathbb{Z}}\) has the initial values \[ F_{-k+2}^{(k)}=\dots=F_{0}^{(k)}=0,\quad F_{1}^{(k)}=1, \] and satisfies the recurrence \[ F_{n}^{(k)}=F_{n-1}^{(k)}+\dots+F_{n-k}^{(k)} \text{ for all } n \in\mathbb{Z}. \] Consider the \(k\)-generalized Fibonacci sequence backward with \(k \geq 3\) be an odd integer, where the characteristic polynomial of this sequence has no dominating zero. Therefore, the application of Baker's method becomes more difficult. Hence, in this paper the authors investigate the coincidence of the absolute values of two terms. The principal theorem gives a lower bound for the difference of two terms (in absolute value) if the larger subscript of the two terms is large enough. A corollary of this theorem makes possible to bound the coincidences in the sequence. In other words, the main theorem and its corollary are as follows: Theorem. Let the sequence \((H_{n}^{(k)})\) be defined by \(H_{n}^{(k)}:=F_{-n}^{(k)}\) with \(n \geq 0\). Assume that \(k \geq 3\) is an odd integer. If \(n>m \geq 0\) then \[ \|H_{n}^k|-|H_{m}^k\|> \frac{|H_{n}^k|}{\exp(7\cdot10^{30}\cdot k^{16}(\log k)^5(\log n)^2)} \] provided \[ n \geq C(k):= 10^{32}\cdot 1.454^{k^{3}}k^{22}(\log k)^5. \] This theorem immediately implies Corollary. Assume that \(k \geq 3\) is an odd integer. Then there is no integer solution \(0 <m<n\) to the equation \[ |H_{n}^k|=|H_{m}^k| \] with \(n>C(k)\).