This repository contains the full theoretical manuscript and the accompanying numerical verification suite for the study of weighted lower-triangular Toeplitz matrices. Fix a ∈ ℂˣ. For each sequence f = (f₀, f₁,. . . , fₙ₋₁) ∈ ℂⁿ, we define the weighted translation-invariant triangular matrix Pₙ⁽ᵃ⁾ (f) by the entry-wise rule Pₙ⁽ᵃ⁾ (f) i, j = aⁱ⁻ʲ fᵢ₋ⱼ for i ≥ j and 0 for i < j. The paper establishes that these matrices form an exact finite-dimensional algebra where the multiplication law is ordinary discrete truncated convolution: (f ∗ g) 𝒹 = ∑ₘ₌₀ᵈ fₘ g𝒹₋ₘ. Inversion is governed by reciprocal formal power series in the ring ℂz/ (zⁿ), allowing for an exact recursion that determines the inverse kernel in O (n²) time. A key contribution of the work is the concept of matrix hybridization via generating-function factorizations F (z) = L (z) U (z). This induces exact matrix products within the same algebra, providing a structured approach to constructing and factorizing complex kernels. The work also identifies a unified multi-parameter rational-binomial family F_ (α, λ) (z) = ∏ⱼ₌₁ᵐ (1 - λⱼ z) ⁻ᵅʲ that is closed under products and inverses, recovering geometric and binomial kernels as special cases. The framework is further extended to block-valued kernels F𝒹 ∈ Mᵣ (ℂ), maintaining the same convolution and inversion formalism. The included software artifacts verify these theoretical results. The script experiments. py reproduces the numerical artifacts, including direct verification of the algebra law, the inverse recursion, and representative semigroup identities in the rational-binomial family. It also provides a compiled single-threaded benchmark exhibiting empirical quadratic scaling for structured inversion compared to dense baselines. The file summary. json provides the hardware and environment metadata for the reported benchmarks.
David Betzer (Thu,) studied this question.