Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
SZALAY, L., A note on the products of the terms of linear recurrences
50 László Szalay Since x m > XQ , it follows that glog k — d m x m e-d* mx m _ e~ckx m e~cx with a suitable constant d * and. c = -f-. m k 'm 4. Proof of the Theorem By Ci,c 2,... we denote positive real numbers which are effectively computable. We may assert, without loss of generality, that the terms of the also holds. Let us observe that it is sufficient to consider the case X{ > no (i = 1, 2,..., is). Otherwise, if we suppose that some x 3 < no (j E {1,2,..., v}) then x m = maxjjxi} cannot be arbitrary large because of the assertion Xj > 6x m. It means that we have finitely many possibilities to choose the i/-tuples (a;! ,..., and the range of Q{x\ ,..., x v ) is finite. So with a fixed d, if inequality (6) is satisfied then q must be bounded. In the sequel we suppose that X{ > n 0 (i = 1,2,..., u). Let ,. .., x„, w, q and s £ S be integers satisfying (6). We may assume that if then ej < q , else a part of 5 can be joined to w q. Using (2), from (6) we have (10) e-t e s= Pi •Pt ( (0 / en) w = (7*r i + — A consequence of the assumptions |7;| > (1 < j < t z) is that (12) whenever X{ —> oo. Hence there exist real constants 0 < S\ ,. .., e v < 1 such that i-l
