Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

GRYTCZUK, A. and VOROBEV, N. T., On some applications of 2 X 2 integral matrices

On some applications of 2x2 integral matrices 39 f 1 n <72m + 2 \ V </2m + l 1; \o 1 J where P

0 = qoiQo = 1, P\ = <7o<7i + 1, Qi = <7i and (9) P

k = qkPk-\ + Pk-2-, Qk = qkQk-i + Qk-2] k = 2n,n = 1,2,... It is easy to see that (8) is true for k = 2. Suppose that (8) is true for some k = 2m. Then we have / Plm-l I

J2m \Q2m-l Ql-m (10) Am-1 + <72 771 + 1-^2771 P2m \ / 1 <72m+2 2 771 V2m y v 0 1 By (9) and (10) it follows that (Pirn1 AmV 1 OWl 92771 + 2 \Q2m-l Q2m/\92m + l 1 / \ 0 X (11) P2

771 ] f 1 <72m+2 Q2771 + I <52m / V 0 1 Denoting the left hand side of (11) by F we obtain (12) F— ( + 1 + 527/1+2^2)71+1 I — I \ Q2771+I ^2m + <72m+2Q2m + l / V P

2 771+1 Pi 771 + 2 Q2771 + I Ö2m + 2 By (12), (11) and (10) it follows that (8) is true for k = 2m + 2, thus by induction (8) is true for every k = 2n, n = 1,2,... Now, we can consider the following product: 1 n'Vi o\" /1 i

x (13) J ^ J J , ->1. Since ("J ÍJ JR-R; - ^ ^ ° xoiy v° 1 / v

1 1/ V

77

1 1 for every positive integer m, then by (13), (14) and (8) for the case k = s — 1 we obtain (15) FQ — [ ^

52 \QS-2 QS- 1