Factorizations of Matrices With Recursive Entries and Related Topics

TL;DR

The paper studies matrices whose entries satisfy a linear recurrence with constants $x,y,z$ by introducing weighted recurrence matrices $P_{\alpha,\beta}^{[x,y,z]}$ and exploring generalized Pascal-type group actions on them. It derives a central unifying factorization (Theorem mainth) that expresses $P_{\alpha,\beta}^{[x,y,z]}$ as a product of a left-weighted Pascal-type block, a middle transformed WRM, and a transpose of a simple Toeplitz block, with explicit recursive formulas for the transformed sequences. This framework recovers many known decompositions (including Tan’s) as special cases and yields determinant formulas that reduce to Toeplitz-determinant computations; it also provides conditions under which the Toeplitz factor becomes diagonal, enabling straightforward determinant evaluations. Overall, the work links recursive-entry matrices to structured factorizations and determinant theory, offering a unified toolkit for factorization and determinant computation in this class of matrices.

Abstract

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.

Theorems & Definitions (19)

• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1: Unifying Factorization
• proof
• Corollary 1
• proof