Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1987. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 18/11)

Bogdan Tropak: Some algebraic properties of linear recurrences

- 69 BOGDAN TROFAK CZIELONA GORA, POLAND) SOME ALGEBRAIC PROPERTIES OF LINEAR RECURRENCES Abstract: In the paper a definition of a form associated to a linear recurrence is given without the restriction that the roots of its characteristic polynomial are different and moreover some properties of this form are studied. This is an extension of some results of P.Kiss C1983.

5 Introduction

. A linear recurrence G = <G V „ of order k(>l) is ^ njn = 0 defined fay rational integers A

4,A

2,...,A and by recursion G^ = Ai

ör,_ 1

+ * • • +

Ak

Gn-k

' n ^ k > where the initial values °o>

Gi>•'>

Gk-i are fixe

d rational integers not all zero, A

kf*«0. To the recurrence G we order a characteristic polynomial g^Cx) as follows Cl> g

G(x)=x

k-A

1x

k"

1-...-A

k_

lx-A

k If a

i

fa

2,...,a

k are the roots of g.Jx) satisfying the condition that a. & a

. for i^j then we define a form f >- J g of k variables X

0,X„

,...,X

k

_

4 by the formula k C2) f fx . ...,X. 1=CdetD> z~

k FI detM. , 0 L o' k ~ lj i=

i t >