Factorizations of Matrices With Recursive Entries and Related Topics

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.

• The paper develops a group of weighted Pascal-type matrices and uses left and right group actions to derive a general factorization for weighted recurrence matrices, unifying several earlier decompositions.
• The factorization reduces determinant evaluation to Toeplitz determinants and yields closed forms for geometric and arithmetic boundary sequences, including powers of y+a+b−ab and 1+y.
• The framework clarifies when the intermediate Toeplitz factor becomes diagonal, while leaving degenerate parameter cases and general Toeplitz determinants as important directions for future research.

Overview

This paper studies infinite matrices $A = [A_{i,j}]_{i,j \geq 0}$ whose entries satisfy the three-term linear recurrence

$$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$

with arbitrary constants $x, y, z$ and initial data prescribed along the first row and column. The authors call these objects weighted recurrence matrices, denoted $P_{\alpha,\beta}^{[x,y,z]}$, where $\alpha$ and $\beta$ are the boundary sequences with a common first term $\alpha_0 = \beta_0$. This single family subsumes several well-studied classes: for $(x,y,z) = (1,0,1)$ one obtains generalized Pascal triangles (and, for constant boundary sequences, the classical Pascal matrix); for $(x,y,z) = (0,x,0)$ one obtains weighted Toeplitz matrices; and for $(x,y,z) = (0,y,z)$ one obtains the so-called $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$0-matrices of Zhong.

The central contribution is a group-theoretic framework: the authors show that a certain family of weighted $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$1-matrices forms a group under multiplication, define two actions of this group (and its transpose) on the class of weighted recurrence matrices, and use these actions to derive a general factorization theorem from which many previously published decompositions follow as special cases. As an application, they obtain closed-form determinant evaluations for several subclasses.

The group of weighted 7-matrices

The key algebraic object is the set $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$2 of $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$3 matrices of the form $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$4, where $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$5, $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$6, and $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$7. These matrices have entries $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$8 and are lower triangular with nonzero diagonal entries $$A_{i,j} = x A_{i,j-1} + y A_{i-1,j-1} + z A_{i-1,j},$$9. The paper establishes the composition law

$x, y, z$0

which immediately yields closure under multiplication, identifies $x, y, z$1 as the identity, and gives the explicit inverse $x, y, z$2. Hence $x, y, z$3 is a group, and since the class is closed under transposition up to the same law, $x, y, z$4 is also a group. This multiplicative structure is the engine behind everything that follows; it is worth noting that the composition law itself was already implicit in Tan's earlier work on biindexed recurrence matrices.

Group actions and the unifying factorization

The paper then proves two action results. First, left multiplication by elements of $x, y, z$5 maps weighted recurrence matrices to weighted recurrence matrices: specifically,

$x, y, z$6

where the transformed first-column sequence is given by the binomial transform $x, y, z$7. The proof is a direct entrywise verification using the defining recurrences of both factors. Second, right multiplication by transposes of elements of $x, y, z$8 has the analogous effect,

$x, y, z$9

with the binomial transform now applied to the first-row sequence $P_{\alpha,\beta}^{[x,y,z]}$0. Using inverses, these yield genuine left and right group actions on $P_{\alpha,\beta}^{[x,y,z]}$1, the family of $P_{\alpha,\beta}^{[x,y,z]}$2 weighted recurrence matrices.

Combining the two actions produces the main result, which the authors call the Unifying Factorization: for any parameters $P_{\alpha,\beta}^{[x,y,z]}$3 with $P_{\alpha,\beta}^{[x,y,z]}$4,

$P_{\alpha,\beta}^{[x,y,z]}$5

where $P_{\alpha,\beta}^{[x,y,z]}$6 and $P_{\alpha,\beta}^{[x,y,z]}$7 are recovered from $P_{\alpha,\beta}^{[x,y,z]}$8 and $P_{\alpha,\beta}^{[x,y,z]}$9 by explicit triangular inversion formulas. The significance of this theorem is its breadth: it contains as special cases the factorization of Moghaddamfar–Pooya for generalized Pascal triangles ($\alpha$0 factored through a Toeplitz matrix), the Edelman–Strang decomposition of the classical Pascal matrix into a product of a lower and an upper triangular Pascal-type matrix, and Tan's fundamental decomposition of $\alpha$1 into two Pascal-type factors surrounding a weighted Toeplitz matrix. In each case the earlier result is obtained by a particular choice of the free parameters $\alpha$2, so the theorem genuinely unifies rather than merely parallels prior work.

A corollary of independent interest refines Tan's decomposition by replacing the weighted Toeplitz factor with an ordinary (unweighted) Toeplitz matrix: whenever $\alpha$3,

$\alpha$4

with $\alpha$5 and $\alpha$6 determined recursively by binomial-transform inversions. Two illustrative examples are worked out explicitly at $\alpha$7: in one case the middle Toeplitz factor collapses to the identity, giving $\alpha$8; in another it collapses to the all-ones matrix $\alpha$9.

Determinant evaluations

The factorization converts determinant computations for weighted recurrence matrices into determinant computations for Toeplitz matrices, which are typically easier. Taking determinants in the corollary above gives the master formula

$\beta$0

using the elementary fact that $\beta$1. All remaining difficulty is concentrated in the central Toeplitz determinant.

The paper characterizes exactly when the middle factor becomes diagonal: this happens if and only if both boundary sequences are geometric, $\beta$2 and $\beta$3. In that case the determinant evaluates in closed form as

$\beta$4

Several classical results follow instantly: the classical Pascal matrix has determinant $\beta$5; the unipotent case $\beta$6 has determinant $\beta$7; and Krattenthaler's evaluation $\beta$8 for constant boundaries is recovered.

For non-diagonal middle factors, the paper computes the transformed sequences explicitly in two further cases. When $\beta$9 and $\alpha_0 = \beta_0$0 with $\alpha_0 = \beta_0$1, induction gives $\alpha_0 = \beta_0$2 and $\alpha_0 = \beta_0$3, whence

$\alpha_0 = \beta_0$4

recovering Krattenthaler's theorem. When $\alpha_0 = \beta_0$5 and $\alpha_0 = \beta_0$6 (arithmetic boundary sequences), the transformed sequences collapse to Kronecker deltas scaled by powers of $\alpha_0 = \beta_0$7, yielding the even-order evaluation

$\alpha_0 = \beta_0$8

Notably, the arithmetic-sequence case is stated only for even orders $\alpha_0 = \beta_0$9; the odd-order determinant is not addressed, and the paper does not explain why the even case is singled out beyond what the computation delivers.

Limitations and open questions

The framework has clear scope restrictions. The Unifying Factorization requires $(x,y,z) = (1,0,1)$0, and the Toeplitz-factorization corollary requires $(x,y,z) = (1,0,1)$1; degenerate parameter regimes fall outside the method. The diagonalization criterion shows that closed-form Toeplitz determinants arise only under restrictive geometric conditions on the boundary sequences, so for general $(x,y,z) = (1,0,1)$2 and $(x,y,z) = (1,0,1)$3 the reduction to a Toeplitz determinant is a simplification but not a complete solution. The determinant results presented are confined to geometric or very simple arithmetic boundary data, and the odd-order analogue of the arithmetic-sequence evaluation is left untreated. Finally, the paper works over $(x,y,z) = (1,0,1)$4 throughout; whether the group-action formalism extends to matrices over general rings, where invertibility of $(x,y,z) = (1,0,1)$5 and $(x,y,z) = (1,0,1)$6 may fail, is not discussed.

Conclusion

The paper organizes a scattered body of factorization results for matrices with recursively defined entries around a single structural principle: a two-parameter group of weighted Pascal-type matrices acting on weighted recurrence matrices from the left and, via transposes, from the right. The resulting Unifying Factorization recovers the decompositions of Edelman–Strang, Moghaddamfar–Pooya, and Tan as parameter choices, and its determinant corollary yields clean closed forms — including $(x,y,z) = (1,0,1)$7 for geometric boundaries — that previously required separate arguments. The main open problem left by the work is the systematic evaluation of the intermediate Toeplitz determinant for boundary sequences beyond the geometric and simplest arithmetic cases.