IJPAM: Volume 20, No. 2 (2005)

NEWTON'S AND SECANT ITERATION OF
QUOTIENTS OF FIBONACCI AND LUCAS NUMBERS

Roman Witu $\l$ a

, Damian S $\l$ ota

$^{1,2}$ Institute of Mathematics
Silesian University of Technology
Kaszubska 23, Gliwice 44-100, POLAND

e-mail: d.slota@polsl.pl

Abstract.The forward and the backward Newton iterations on the elements $\mathbb{X}_{n+1}/\mathbb{X}_n$ , where $\mathbb{X}_n=A\, \alpha^n +B\, \beta^n$ , $\alpha :=(1+\sqrt{5})/2$ , $\beta :=(1-\sqrt{5})/2$ are studied. Moreover, some relations for the sequence $G(\mathbb{X}_{n+1}/\mathbb{X}_n,$ $\mathbb{Y}_{n+1}/\mathbb{Y}_n)$ are studied, where , $\mathbb{X}_{n}$ and $\mathbb{Y}_n$ are recurrence sequences of the above-mentioned type. For example, it is proved that the identities $G\Big(\frac{F_n}{F_{n-1}},\frac{F_m}{F_{m-1}}\Big) = G\Big(\frac{L_n}{L_{n-1}},\frac{L_m}{L_{m-1}}\Big) = \frac{F_{n+m-1}}{F_{n+m-2}}$ and $G\Big(\frac{L_n}{L_{n-1}},\frac{F_m}{F_{m-1}}\Big) =\frac{L_{n+m-1}}{L_{n+m-2}}$ are characteristic for Fibonacci and Lucas numbers.