一些迭代递归公式

Some Iterative Recurrence Formulas

Fine提出如下两个猜想。猜想(Ⅰ_b):设0<b<1,a是bx^2+x-1=0的正根(0<a<1)函数G如下递归定义G(0)=0,G(n)=n-[bG(G(n-1))](n≥1)则D_n=G(n)-an有界。猜想(Ⅱ_k):设k≥2,a为x^k+x-1=0的唯一正根,0<a<1,函数G定义为G(0)=0,G(n)=n-G_k(n-1)(n≥1),其中G_k(n)=G(G_(k-1)(n)),则D_n=G(n)-an有界。本文证明了:(1)猜想(Ⅰ_b)正确,并且(-2ba^2)/(1-a^2b^2)≤D_n≤(2a)/(1-a^2b^2);(2)当k=2,3,4时,猜想(Ⅱ_k)正确。而k≥5时则不正确。

Fine raised the following two conjectures. conjecture (I_1)~2 Let 0<b<1. a the unique positive root of bx^+x-1 = 0 (0<a<1). The function G is defined by G(0) = 0. G(n) = n- [bG(G(n- 1))] (n≥1). Then D_n= G(n) -an is bounded. Conjecture (II_1)_2 Lef k≥2. The function G is defined by G(0) = 0, G(n) = n-G+k(n- 1) (n≥1), where G_k(n) = G(G_(k-1) (n)).Then D_n= G(n) - an is bounded. In this paper we proved that (1) the conjecture (I_b) is true and-2ba^2/(1-a^2b^2)≤D_n≤2a/(1-a^2bk^2), (2) the conjecture (II_k) is true for k=2,3,4,and false for k≥5