Certain matrices related to Fibonacci sequence having recursive entries

Moghaddamfar, Ali Reza ; Salehy, S.Navid ; Salehy, S.Nima

The electronic journal of linear algebra, Tome 17 (2008), pp. 543-576.

Résumé

Summary: Let $\varphi = (\varphi i )$ i$\geq 1$ and $\psi = (\psi i )$ i$\geq 1$ be two arbitrary sequences with $\varphi 1 = \psi 1$ . Let A $\varphi ,\psi ( n)$ denote the matrix of order n with entries a i,j , $1 \leq i$, j $\leq n$, where a 1,$j = \varphi j$ and a i,$1 = \psi i$ for $1 \leq i \leq n$, and where a i,j = a i - 1,j - 1 + a i - 1,j for $2 \leq i$, j $\leq n$. It is of interest to evaluate the determinant of A $\varphi ,\psi ( n)$, where one of the sequences $\varphi $or $\psi $is the Fibonacci sequence (i.e., 1 , 1, 2, 3, 5, 8, . . .) and the other is one of the following sequences: $\alpha (k)$ = ( k - times TH ß 1 , 1, . . . , 1, 0, 0, 0, . . .) , $\chi (k)$ = (1 k , 2 k , 3 k , . . . , i k , . . .), $\xi (k)$ = (1 , k, k 2 , . . . , k i - 1 , . . .) (a geometric sequence), $\gamma (k)$ = (1 , 1 + k, 1 + 2k, . . . , 1 + (i - 1)k, . . .) (an arithmetic sequence).

Détail

Classification : 15A09, 11B39
Keywords: inverse matrix, determinant, LU-factorization, Fibonacci sequence, generalized pascal triangle, recursive relation