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

- 75 3. A connection between gCx) and F . g Lemma 2

. Let gCx) = x

k - A x

k ~ 1 - ... - A, x - A, , i k -1 k * u(x) = x° - B x'

3"

1 - B x — B , 1 a -1 a ' v(x) = x

r - G x

r ~

1 - ... - G X — C 1 r -1 r and let gCx) = u(x) v(x)

. Fg [

Xo'

' * * *

Xk~i} is the associa

te

d form to gCx) then F 9 where F^ and F^ are forms associated to u(x) and vCx), respectively and r Zj =-IC

r.

tX

j+

l, j=0,l,

. . . ,s~l with C

0=-l, t = o e = - 2 B

a.

nX. +

n, i=0,l,...,r-l with B

0=-l. n = 0 Proo

f: For the brevity put al = ~ Ak-l > 1 * 1 * k > bn - " B

s-n > 1 ^ n ^ s , c = - C , 1 £ m <1 r ín r -m * and let a = <x be a root of uCx). By C35 and C4) we have k k k z

a = 2 S

t<«>X

t_

t = 2 X

t_

l 2 a,«

1"

1 = t =i t =i I=t k k = 2 X. , 2 2 c b a

m+

n"

1 = <

6- t -l m n t =1 I =t m + n = t O^rn^r O^n^B