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 7 Proof. It is a result of A. Baker [1]. (We mention that this result was improved by A. Baker and G. Wüstholz [2], but we do not calculate the exact values of the constants thus we use only the result of Lemma 1.) Lemma 2. Let j be a real number with 0 < 7 < 1, ,x

2,...,x

m be an integer defined by (7) and G^ (1 < i < m, 2 < m) be sequences defined by (4), that is, for any 1 < i < m the polynomial has a dominant root a; = a^'. If Xi > 7•inax(xi

, x'2, • • •, x

m), then there are effectively computable positive constans Co and ??

2 depending only on the sequences G'

(i

) and 7, such that where < e C2

X for any x ~ max(xi, x-t. ..., x

m) > ni. Proof. For the proof see Lemma 2 in [9]. After these lemmas we present the proofs of the theorems. We mention that the constants c; and n

?- (i > 2) shall always denote effectively computable positive real numbers depending 011 7, the primes pi,p2, • • • ,Pr and the parameters of the recurrences. One can compute their explicit values similarly as in [9-10]. Proof of Theorem 1. Suppose that the conditions of the theorem are fulfiled and (12) with a suitable constant c'

Q > 0 and sufficiently large x. By Lemma 2, (13) where |

ít

| < e cx if x > no- On the other hand, by (9b) m I log + x> I

oS Y1 lo

g Y1 l

oS l^'l (14) = 1 > e

i=

1 i=i if X > 77

.3. Using (13) and (14), from (12) we can get the inequalities s (1+0 < < e(

c'oc3>" — e~

c*

x rn m n <».•«* n I ai

av i1