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

- 70 where D = 1 2 M = Xo 1 . 1 Xk-t«i k -1 k — 1 ' k - 1 " k — 1 .a ~a. * . . . af 1 v - 1 x. + 4 k From C2) it- follows that Tor k>2 the restriction on the roots of g

GCx) is essential. P. Kiss C1983) has studied the form f and from it. he has derived some properties of linear recurrences. In this paper we define for arbitrary linear recurrence G a form F such that if the roots of gCx) are different 9 then F — f .Further we show that some results of P.Kiss g g remain valid in this general case. Finally we prove a connection between the factorisation of gCx) and of F 2. Definition and properties of F . £L Let G be a linear recurrence of order k and let gCx) = x

k-A

1x

k~

1-.

..-A

k

lx-A

k , A

k * 0 , be its characteristic polynomial. Define for 1=1,2,...,k k C3) g

LCx> = - I A

k

mx

TO_

1 with A

0=-l k -m m = l and for the variables X

Q,X^,. . . , X

k _

± k C4) «

a = 2 e.Ca^^ l=i where a is a root of g<x) .