Application of the \(\varepsilon\)-algorithm to the ratios of \(r\)-generalized Fibonacci sequences (Q2721667)

For \(r\geq 2\) an \(r\)-generalized Fibonacci sequence \((V_n)_{n\geq 0}\) is defined by the linear recurrence of order \(r\), \(V_n= a_0V_n + a_1V_{n-1}+\cdots +a_{r-1}V_{n-r+1}\), \(n\geq r-1\), where \(a_0, a_1,\dots , a_{r-1}\) are complex numbers, \(a_{r-1}\neq 0\) and \(V_0,V_1,\dots ,V_{r-1}\) are given values. If the limit \(q=\lim_{n\to \infty} \frac{V_n}{V_{n+1}}\) exists, then \(q\) is a root of the characteristic equation \(x^r= a_0x^{r-1} + \cdots +a_{r-2}x+a_{r-1}\). Therefore, the sequence \((V_n)_{n\geq 1}\) can be used for approximation of roots of algebraic equations. The authors apply the so-called \(\varepsilon\)-algorithm, which is a generalization of the method of Aitken, see \textit{P. Henrici} [Elements of numerical analysis (New York, Wiley \& Sons) (1964; Zbl 0149.10901)] to accelerate the convergence of the sequence \((\frac{V_n}{V_{n+1}})_{n\geq 0}\). This extends an idea of \textit{J. H. McCabe} and \textit{G. M. Phillips} [Fibonacci numbers and their applications 1, 181-184 (1986; Zbl 0589.10010)].