Journal of the
Korean Mathematical Society JKMS

The notion of the generalized Fibonacci matrix $\mathcal {F}_n^{(a,b,s)}$ of type $s$, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper [23]. Regular case $s=0$ is investigated in [23]. In the present article we consider singular case $s=-1$. Pseudoinverse of the generalized Fibonacci matrix $\mathcal {F}_n^{(a,b,-1)}$ is derived. Correlations between the matrix $\mathcal {F}_n^{(a,b,-1)}$ and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.

Keywords: generalized Fibonaci numbers, generalized Fibonaci matrix, Lucas numbers, Lucas matrix

MSC numbers: 05A10, 11B39, 15A09