Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

H.-MOLNÁR, S., Approximation by quotients of terms of second order linear recursive sequences of integers

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 21-26 APPROXIMATION BY QUOTIENTS OF TERMS OF SECOND ORDER LINEAR RECURSIVE SEQUENCES OF INTEGERS Sándor H.-Molnár (Budapest, Hungary) Abstract. In the paper real quadratic algebraic numbers are approximated by the quotients of terms of appropriate second order recurrences of integers. AMS Classification Number: 11J68, 11B39 Keywords: Linear recurrences, approximation, quality of approximation. 1. Introduction Let G = G{A, B,Gq,G\) — {Gnj^-o be a second order linear recursive sequence of rational integers defined by recursion G

n = AG

n1 + BG

n2 (

n > 1) where A, B and the initial terms Go, Gi are fixed integers with restrictions AB 0, D — A

2 + 4B ^ 0 and not both Go and G'i are zero. It is well-known that the terms of G can be written in form (1) G

N = CIA

n-bß

n, where a and ß are the roots of the characteristic polynomial x

2 — Ax — B of the sequence G and a = G

l~^

ß

o

ß

, b = (see e. g. [7], p. 91). Throughout this paper we assume |a| > \ß\ and the sequence is nondegenerate, i. e. a/ß is not a root of unity and cib / 0. We may also suppose that G

n / 0 for n > 0 since in [1] it was proved that a non-degenerate sequence G has at most one zero term and after a movement of indices this condition can be fulfilled. In the case D = A

2 + 4B > 0 the roots of the characteristic polynomial are real, \a\ > \ß\,(ß/a)

n — 0 as n oo and so by (1) Hm %±l - a follows [61. n—*oo u n In [2] and [3] the quality of the approximation of a by quotients G

n+i/G

n was considered. In [3] it was proved that if G is a non-degenerate second order linear recurrence with D > 0, and c and k are positive real numbers, then G'n + i a — G'n 1 < I G

N I ^