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

- 73 the coefficients of (^o*

* * * *^k-1J ar

s rational integers. v Moreover g

kCx) - 1 hence the coefficient of l is equal to k n g

k (a) = 1 . i = 1 For the proof of second part of the theorem put g

0CxD=gCx3 and remark that g (x)+A g (x) = -i—i k 1+

1 for 1=1, 2,..., k . x Now for a. = a

. , l^j^k and for any n£G we have k k « 2 G

n+11 = I k^co+A^J G

n+

l_

t = 1=1 1=1 k k = fg CcO+A, + 5 g, CcOG , + 5 A, , G , kjn I - 1 r> +1 - 1 k-l+1 n+l-1 I =2 1=2 k-1 k-1 k-1 = A, G + 5 A. . G Jt

l + 2 g, CcOG . = ^ A . Q + k n k-Lr> + L n + l k-ln-t-L 1=1 I =1 k-l I = O k-l k-l k + 2 g

lCo5G

n + l = G

n+

k - 2

g

l(«)G

n + l = 2

S

lCa>G

n + l 1=1 1=1 L =1 because G , = G , g, Ca) . n+k n+k k From the above calculations we obtain 2 S

LCa.)G

n+

l l=i = 0 i. 5 g, (a. )G x. bl t n +1 - 1 1=1 X — F Í

g ,...,G . 1 Í] ex. = g^n' * n+k-ij ** t = F fű ,G . ]c-l>

k_

iA

u g^ n* r> +1 * n + k-1 J k and the proof easily follows by the induction.