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
40 F. Mátyás Recently, some papers have been pnblicated in this field. (For more information about the list of these papers see [1].) One of the latest such papers was written by Nathan D. Cahill and Darren A. Narayan [1]. They have constructed such family of tridiagonal matrix determinants of k x k which generate any arbitrary linear subsequence Fak+ß or L ak+ß (k = 1,2,...) of the Fibonacci or Lucas numbers. For example, /1 0 0 8 1 Fik-2 = det, 1 7/ The aim of this note is to investigate suitable matrix determinants of n x n which form the terms G n of the Fibonacci-type sequences defined by (1). In this paper we suppose that in (1) p\ ^ 0, jt>j = 0 (2 < j < Ar — 1 for 3 < k),pk = ±1, and G\ ^ 0, that is we deal with the family of sequences (2) Gn = G n (pi, 0,..., 0, ±1, G 2-k, Ö3-/C, . •., Gl) . (Naturally, the sign ± in (2) is fixed in a given sequence.) For our aim we construct the matrix A nx n — (atj) of complex numbers by the following forms: ci^i = Gi, = —e J+ lGj-k (2 < j < k ), Oj+i,j - —e 3 (1 < j < n — 1), a,j tk+j-1 = —e k+ i (2 < j < n + 1 — k),ajj = pi (2 < j < n) and the other entries are equal to 0. That is, (3) A Jl Xn I G\ -e 3G-2-k -e 4G 3k - -e k+ 1G 0 0 0 ... 0 0 \ -e 3 P l 0 - 0 -e k+ 1 0 ... 0 0 0 -e 3 pi - 0 0 -e fc+ 1 - 0 0 V 0 0 0 ... 0 0 0 ... -e 3 P lJ where e — — \ if p^ = — 1 and e = —i if pk = 1.
