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

A note on the products of the terms of linear recurrences 51 and V V eillN" <

sw

q < c2 N W1=1 1=1 As before, leta; = a;i+'-- + a:

I

/ and applying (9) we may write log Cl + X log \~j

v I < log s + q log w < log c

2 + x log ^ |

. Since log s > 0, we have (13) logc

3 + x log \j

u\ < qlogw < log c

2 + zlog |7i| with c

3 = From (13) it follows that (14) X x c

4 - < log w < c

5 q q with some positive constants c

4, C5. Ordering the equality (11) and taking logarithms, by the definition of £{ we obtain Q = log sw dUi=i killn logfl 2 = 1 ai \ H ) < 1=1 i1 where Q / 0 if we assume, that x

l > n

0 for every i — 1, 2,..., v, and c* is a suitable positive constant (i — 1,2,

..., v). Applying Lemma 2 and using the notation x = £1 + • • • + x

v, it yields that (15) On the other hand Q < e~

c6(

xi+---+

x^) — e~

Ce,

x

. (16) Q = log 5 + q log w - log d - log JJ \a,i\ — xi log I711 x

v\og\iv\ 1=1 where logs = ei.logp! + j- e

t\ogp

t (see (10)). Now we may use Lemma 1 with 7T

r = w = M

r, since the ordinary heights of pj (j = 1,2,...,/), nr

=

1 K'l and |7i| (i ~ 1, 2,.. . ,*/) are constants. So i?' = q. In comparison