Summary: Consider the sequences \(\{u_ n\}\) and \(\{v_ n\}\) generated by \(u_{n+1}= pu_ n- qu_{n-1}\) and \(v_{n+1}= pv_ n- qv_{n-1}\), \(n\geq 1\), where \(u_ 0=0\), \(u_ 1=1\), \(v_ 0=2\), \(v_ 1=p\), and \(p\) and \(q\) real and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of \(\{u_ n\}\) and \(\{v_ n\}\), respectively. Define \(r_ n= u_{n+d}/u_ n\), \(R_ n= v_{n+d}/v_ n\), where \(d\) is a positive integer. J. H. McCabe and G. M. Phillips [Math. Comput. 45, 553-558 (1985; Zbl 0581.10005)] showed that for \(d=1\), applying one step of Aitken acceleration to any appropriate triple of elements of \(\{r_ n\}\) yields another element of \(\{r_ n\}\). They also proved for \(d=1\) that if a step of the Newton-Raphson method or the secant method is applied to elements of \(\{r_ n\}\) in solving the characteristic equation \(x^ 2-px+q=0\), then the result is an element of \(\{r_ n\}\).
The above results are obtained for \(d>1\). (For the ordinary Fibonacci numbers this was considered by M. J. Jamieson [Am. Math. Mon. 97, 829-831 (1990; Zbl 0735.11007)].) It is shown that if any of the above methods is applied to elements of \(\{R_ n\}\), then the result is an element of \(\{r_ n\}\). The application of certain higher-order iterative procedures, such as Halley’s method, to elements of \(\{r_ n\}\) and \(\{R_ n\}\) is also investigated.