Some Properties of a Class of Band Matrices

Let $A(2r + 1,n)$ denote the $n \times n$ band matrix, of bandwidth $2r + 1$, with the binomial coefficients in the expansion of ${(x - 1)^{2r}}$ as the elements in each row and column. Using the fact that the rows of $A(2r + 1,n)$ provide the coefficients for the $2r$th central difference, a number of properties of $A(2r + 1,n)$ are obtained for all positive integers $r$ and $n$. These include obtaining explicit formulas for $\det A(2r + 1,n),{A^{ - 1}}(2r + 1,n),||{A^{ - 1}}(2r + 1,n)|{|_\infty }$ and an upper triangular matrix $U$ such that $A(2r + 1,n)U$ is lower triangular.