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

- 71 Let a ,a ,...,ct

k be roots of gCx) Cobviously a root of multiplicity r is taken r times) and X

Q,X

±,...,_

t be variables. The form k i -1 i will be called a form associated to gCx) CS) F 9 Lemma 1. If gCx) is a polynomial having distinct, root« then F = f . Proo

f: Assume that the degree of gCx) is k and er , 1 i k are its roots. Consider the following system of equations C6) y» + y

2 + . . + y

k = Xo + a

2y

2 + . . + — x 1 + <y

2 + . . + = x

2 Í

_

1y + i + . . + = Xk-x with y's as unknowns. By the assumption of lemma

, C6) is Cramer's system hence C7) y = CdetD)

_

1detM. C-1)

Lfor i=l,2,...,k where D and M. are as in C2) i On the other hand it is easy to verify that for a. . = -J t » J £ g

;Ca. ) Irzrj ' i ^ ^ k we have D~

1 = [a. . J Therefore from C

6) we obtain that k C8) 1 K a. = 2 ai.i

Xi-i = g^TcTT 2 g

lCa.)X

l_

i = px-T • 1=1 1 I =1 1