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

- 74 Theorem 2

. (see Thm. 2 in Kiss, 1983.) If = a + a„ a. +•...+ a. „ a

k_

1, n^O t , n O, n 1, r> i k — 1, r» i where k -t -i a. = Q .. , . - 5 A.G . . , OStSk-1 t , ri ri + k-t-1 j n + k-t-j-1 * j = 1 and if k U a n j*. then u = U-i^-'AS U . n l kJ o Proo

f: For 1 S i 5 k we have k k k k -rn z = IX 5 Í—A , a

1 ~

m) = - 2 X « 2 . a

1 •» a m- i ^ k-I l J m-1 k 1 -in v m=l I =m m =1 I =0 k-lk-l k-l f k-l-1 ^ = - 2 K , x «, = I . I K , x « a

l t k-l-roro-l Ok-l-1 k-l -m m-l I x. I =0 m = 1 1=0^ rn= 1 k-l r k-l-1 2 a

1 X. , - 2 AX., t |k-l-l mk-l-m-lj I =0 ^ m= 1 ^ putting X^ = Q

n+

r, r=0,

1,...,k—

1 we obtain and k-l f k-l-1 ^ z = 5 a

1 G _ , „ - 2 A G _ . = a. I n + k ~

1 -1 *- j n + k-l - j -1 I , n I =0 ^ j =1 J and now by definition of F^ and by Theorem 1 we get the proof.