The \(k\)-Fibonacci matrix and the Pascal matrix
The classical Fibonacci matrix is extended to the concept of the \(k\)-Fibonacci matrix, defined as a unipotent lower triangular \(n\times n\) Toeplitz matrix \({\mathcal F}_n(k)=[f_{i,j}]_{i,j=1,\dots,n}\) with entries \(f_{i,j}=F_{k,i-j+1}\) if \(i\geq j\), 0 otherwise, where the \(k\)-Fibonacci numbers \(F_{k,n}, n\in{\mathbb N}\) verify the reccurence relations \[ F_{k,n+1}=kF_{k,n}+F_{k,n-1},\;n\geq 1, \] with the initial conditions \(F_{k,0}=0,\;F_{k,1}=1\). Two factorizations of the Pascal matrices are obtained, which involve the \(k\)-Fibonacci matrices and two types of matrices. Some combinatorial formulas containing the \(k\)-Fibonacci numbers are derived and the product of a \(k\)-Fibonacci matrix and its transpose is studied.
scientific article; zbMATH DE number 5987910