Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

ZAY, B., A generalization of an approximation problem concerning linear recurrences

A generalization of an approximation problem concerning linear recurrences BÉLA ZAY* Abstract. Let {£?„} be a linear recursive sequence of order i(>2) defined by G

n = A\G

n-i-\ hA

tG

n-t for n>t, where A

x,...,A

t and G

0,...,G

t-1 are given rational integers. Denote by a^ ,a2,...,a

( the roots of the polynomial x

t — Aix

t~

x A

t and suppose that |c*i|>|a,| for 2<i<t. It is known that lim °q + s — o^, where s is a positive integer. n—* oo n The quality of the approximation of a

x by rational numbers in the case was investigated in several papers. Extending the earlier results we show that the inequality holds for infinitely many positve integers n with some constant c if and only if Let be a A;

T

H order (k > 2) linear recursive sequence defined by G

n ~ AiG

n-i + A

2G

n-2 H + A

kG

nk for n > k, where Ai,

..., Afc, and G\,

.

.., Gk are given rational integers with A

k ^ 0 and G

2 + • • • + G\_

x ^ 0. Denote by «i,

..., a

t the distinct roots of the characteristic polinomial f{x) = x

k- Aix

k~

l A

k = {x- a

i)

m*(x - a

2)

m

2 • • • (x - a

t)

m<

. Using the well known explicite form of the terms of linear recursive sequences, G

n can be expressed by t ( mi \ t « = E E an

n

i~

l K = E (" ^ °) 2 = 1 \j= 1 J i= 1 x Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 16975 and 020295.