Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

MÁTYÁS, F., Genaralize d Fibonacci-type numbers as matrix determinants

42 F. Mátyás Now, we shall deal with the case n > k. If n = k + 1 then 3G- • -<=

4 Pi det (A

fc+lxfc+

1) = det / G1 -e

óG

2-k -e

4G

3-k 3 o Pi 0 \ 0 -e

k+

1G

0 0 \ 0 -c*+

1 0 0 —e" Pi = PiG

k + e

3 det / Gi -e

3G

2_* -e

4G

3_

f

c -e

3 pi 0 0 -e

3 pi \ 0 0 0 -e

kG1 ü \ 0 0 —e

; 0 0 / Developing successively the resulting determinants with respect to their last rows, we have det (A

nx

n) = PiG

k + (e

3)"1 det ( ^ _J

+

1 ) = PiG

k - e

3

k~

3e

k+

lGi = PiG

k + p

kG

x = G

k+

1. Let us suppose that det (A

n_

jX

n_j) = G

n-j (1 < j < k) holds for an integer n > k + 2. In this case det (A„

x„) = det / Gi -e

c 0 V 0 r2 — k = piGn-i + e

3 det Pi (Gi —e

v 0 V o •

k+

lG

0 0 0 0 _

ek+

1 0 u 0 _

ek+l '2-k Pi —e" 0 0 0 Gq -e*

+

1Gn 0 0 0 0 0 \ 0 0 0 0 -e

3 pi J 0 0\ 0 0 0 0 0/ Now, develop successively the resulting determinants with respect to their last rows. Then one can get the following equalities: k1 det (A

n

x„) = piG

n_i + (e

3) (~e

k+

l) G

n— piG

n-i - e

2G

n-k — PiG>j_i + p

kG

nk — G

n. This completes the proof of the Theorem.