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

8 P. Kiss & F. Mátyás if Cq < C3. From this estimation, with < e" (15) n i=i < 1 + lel + e~

cr

x < 1 + e" follows if x > 77.4, which implies that \s\ < (l + eCs

I)[lK'| < (l + e~

c

5'

c) i=

1 ^logja.l+a: ^ log I a

, I < if x > n

5. Since by (5), using the notation y = max (e

;), l<i<r e

Ce

X > |s| = Y[pt > II

2

6' ^ ^ = erl0g

; therefore (16) Let A = i = l i'=l y = max (e,;

) < cjx. 1 < I < r log . It is clear by (9a) that A ^ 0, while by (15) and the Et 1=1 properties of the logarithm function, (17) 0 < A < log (l + e~

C5

X) < e Now we give a lower estimation for A = ei l

o8' Pi - Io

g \

U

i 1 ~ zL Xi lo

g i1 1=1 Since the numbers pi, |cí

;| and \cn

[ are algebraic ones with bounded heights, further 011 the numbers e; and X{ are bounded above by c-jx (see (16)) and x, respectively, thus by Lemma 1 (18) A > ec® l

oS'

c. (17) and (18) imply that c

5x < c

8 log x,