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 LÁSZLÓ SZALAY Abstract. For an integer u>l let (t=l,...,t/) be linear recurrences defined by GM=A[

i

)G™

l + ---+Al

i>G

nk

i (n>ki). In the paper we show that the equation dG

(

x\

)---G

(

x

1

/J=sw

q

, where d,s,w,q,x, are positive integers satisfying some conditions, implies the inequality q<qo with some effectively computable constant q

0

• This result generalizes some earlier results of Kiss, Pethő, Shorey and Stewart. 1. Introduction Let GW = {G

(n

]}™

= o = 1,2,. .., z/) be linear recurrences of order k{ (.ki > 2) defined by (1) Gg) = A^G^ + • - • + A%G<tl

k

i (n > ki), where the initial values G^ (j = 0,1,..., k

z — 1) and the coefficients (I = 1,2, ...,ki) of the sequences are rational integers. We suppose, that / 0 and there is at least one non-zero initial value for any recurrences. By a^ = 7i, a^

,..., a^ we denote the distinct roots of the characteristic polynomial Pi(x) = x

k

i - A[^x

k'~

l of the sequence and we assume that t{ > 1 and |7;| > for j > 1. Consequently |7,| > 1. Suppose that the multiplicity of the roots 7; are 1. Then the terms of the sequences G^ (i = 1,2,.. ., v) can be written in the form