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

Kiss, P. & MÁTYÁS, F., On products and sums of the terms of linear recurrences

On products and sums of the terms of linear recurrences 5 GÍ

1» - , P. Kiss [6] proved that if G'^

1' is defined by (4) then, under some conditions, > e

c x for all integers s £ S and x > nwhere c' and n' are effectively computable positive constants depending only on the pimes pi,p2, • • •,p

r and G'

1'. P. Kiss gave a summation of the results concerning this topic in [7], where among others there were cited two theorems (Theorem 3 and Theorem 6) without proofs hoping that the paper containing the proofs had already appeared. Unfortunately, because of some technical reasons, these proofs can appear only in this paper. So the purpose of this paper is to restate the above theorems and to present their proofs. These theorems generalize and extend the result of P. Kiss [6] for the products (and the sums) of terms of linear recurrences defined by (3) and (4). 2. Results For brevity we introduce the following abbreviations: 711 (7) n,,.,,, *

m =n

G£ 8 = 1 111 (8) = i=i where a,*i, X2, •.., x

m are positive integers. The following two theorems will be proved. Theorem 1. Let 7 be a real number with Ü < 7 < 1 and let S be the set of integers defined by (5). Suppose that for any 1 < i < rn the polynomial </'' (x) defined by (2) has a dominant root cvj = cv^' and the sequence G'

2' is defined by (4). Then there exist positive real numbers cq and no such that if x = max (x

l

) > no, l<i<m m (9a and 96) TT ^ S and X{ > jx for 1 < i < m, i-l then (10) \s -n*

l,*>,...,xJ>e

eo

X for any s £ S and positive integers xi,x2,...,x

m. The constants cq and uq are effectively computable positive numbers depending only 011 7, the primes Pi,P2, • • • ,Pr and the parameters of the sequences G

(l

) (1 < 1 < rn).