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

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 39-^3 GENERALIZED FIBONACCI-TYPE NUMBERS AS MATRIX DETERMINANTS Ferenc Mátyás (Eger, Hungary) Abstract. In this note we construct such matrix determinants of complex entries which are equal to the numbers defined by Fibonacci-type linear recursions of order k>2. AMS Classification Number: 11B39, 11C20 1. Introduction Let. k > 2 be an integer. The recursive sequence {G

n}^_

2-A; or

(l

e

r k is defined for every n > 2 by the recursion ( 1) G

n = piG

n1 + P'2G

n_2 H f p

kG

n-k, where p

t (1 < i < k) and Gj (2 — k < j < 1) are given complex numbers and PiPkGi is not equal to zero. For brevity, we will use the formula G

n = G

n (pi,P2,

..., pfe, Gz-k, G$-k, • •., G\

), as well. In the case k = 2 we get the wellknown family of second order linear recurrences of complex numbers. The two most important sequences from this family are the Fibonacci {F

n} and the Lucas {L

n

} sequences, where F

n = G

n{ 1,1,0,1) and L

n = G

n( 1, 1,2,1), respectively. The close connections between the Fibinacci (and Lucas) numbers and suitable matrix determinants have been known for ages. For example, it is known that for k > 1 Fk is equal to the following tridiagonal matrix determinant of k x k: \ 1/ /1 Fk = det