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

- 72 Since k kCk-O n g'[a.j = C—

1) 2 .CdetD)

2 then from C2), C7), CS) and C5!> we get k fg(

Xo" •• '

Xk-l]'

CdetD>

2"

k " [

C1>i_

1yi det

D) " kCk-l

) k k = CdetD>

zC—

1) 2 n

y . n. - F [x

o

> ...,x

k_

t) . 1=1 V = 1 I. This ends the proof. Theorem 1

. Ccomp. Thm.

1 in Kiss, 1083), The form (^o'

" * * '

Xk-1J ha

s rational integer coefficients and the coefficient of x

k_i i

s one. Furthermore for all integer n^O, where F

Q= F

g (g

q,G

±, . . . ,

G

k

_

4J . Proof: By C3) and C4> we can write k k -1 2 g, Ccc. > X. = 2 u ö I i l-l m l I =1 where u = u X^, . . . ,X, , are linear forms with m m ^ O k-lj rational integer coefficients and then k F 9 (

xo>--'

xk-J - n L = 1 k -1 2 u oT TO V TO — O and the coefficients of u . . u. k

„

1 are rational as O k - 1 symmetrical polynomials in . Since a. 's are algebraic integers then these coefficients and in particulary