Evaluating some determinants of matrices with recursive entries
Bull. Korean Math. Soc. 2009 Vol. 46, No. 2, 331-346
Printed March 1, 2009
Ali Reza Moghaddamfar, Seyyed Navid Salehy, and Seyyed Nima Salehy
K. N. Toosi University of Technology
Abstract : Let $\alpha=(\alpha_1, \alpha_2, \ldots)$ and $\beta=(\beta_1, \beta_2, \ldots)$ be two sequences with $\alpha_1=\beta_1$ and $k$ and $n$ be natural numbers. We denote by $A_{\alpha,\beta}^{(k,\pm)}(n)$ the matrix of order $n$ with coefficients $a_{i,j}$ by setting $a_{1,i}=\alpha_i$, $a_{i,1}=\beta_i$ for $1\leq i\leq n$ and $$a_{i,j}=\left \{ \begin{array}{lll} a_{i-1,j-1}+a_{i-1,j} & \mbox{if}& j\equiv 2, 3, 4, \ldots, k+1\ \pmod{2k}\\ a_{i-1,j-1}-a_{i-1,j} & \mbox{if}& j\equiv k+2,\ldots, 2k+1 \pmod{2k} \\ \end{array} \right. $$ for $2\leq i,j\leq n$. The aim of this paper is to study the determinants of such matrices related to certain sequences $\alpha$ and $\beta$, and some natural numbers $k$.
Keywords : determinant, LU-factorization, recurrence relation
MSC numbers : 11C20, 15A15, 15A36, 15A57
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