Factorizations of Matrices With Recursive Entries and Related Topics

This article examines matrices whose entries are determined by recursive relations of the form $A_{i,j}=x{A}_{i,j-1}+y{A}_{i-1,j-1}+z{A}_{i-1,j}$, where $x,y,z$ are constants, and the initial conditions are defined along the first row and column. We present a general decomposition for such matrices and show that many of the known decompositions are particular cases of this more general decomposition. Additionally, we provide a decomposition of these matrices into Pascal-like matrices and a basic Toeplitz matrix.