David Betzer

Shared-Factor Kronecker-Sum LU and Tensor-Product Structured Matrices

This paper establishes an exact LU theory for finite low-separation-rank Kronecker sums in arbitrary tensor dimension. The principal theorem treats matrices of the form M = ∑ᵣᵥ₌₁ ⊗ᵈₛ₌₁ Aᵥ,ₛ, where each mode shares fixed unit-triangular outer factors, Aᵥ,ₛ = LₛTᵥ,ₛUₛ, and the term...

Prime-Pair Incidence Matrices: Almost-All Higher Prime Correlations and Pascal Transforms

This paper studies the binary prime-pair incidence matrix defined by (Aₙ)ₘ,ᵣ = 1ℙ(m − r)1ℙ(m + r), whose row index records the midpoint of a prime pair and whose column index records its half-gap. Rows therefore encode unequal Goldbach representations, while columns count prime p...

Shared-Factor Kronecker-Sum LU and Tensor-Product Structured Matrices

This paper establishes an exact LU theory for finite low-separation-rank Kronecker sums in arbitrary tensor dimension. The principal theorem treats matrices of the form M = ∑ᵣᵥ₌₁ ⊗ᵈₛ₌₁ Aᵥ,ₛ, where each mode shares fixed unit-triangular outer factors, Aᵥ,ₛ = LₛTᵥ,ₛUₛ, and the term...

The c=2 q-Carlitz Smith Form and q-Pascal Matrix Structures

This paper solves the complete c = 2 case of the Kuperberg–Conlon q-Carlitz Smith-form problem over the Laurent polynomial ring ℤ[q,q⁻¹]. For every a,b ≥ 2, the q-Carlitz matrix C(a,b,2;q) is proved to be unimodularly equivalent over ℤ[q,q⁻¹] to an explicit two-factor Smith form...

Hankel Matrices of Linear Recurrences: Unimodular Congruence and Pell Applications

This paper develops a general unimodular reduction theory for Hankel matrices generated by scalar linear recurrences. Its principal theorem shows that every sufficiently large square Hankel section arising from a recurrence of order d is unimodularly congruent, over an arbitrary...

The c=2 q-Carlitz Smith Form and q-Pascal Matrix Structures

This paper solves the complete c = 2 case of the Kuperberg–Conlon q-Carlitz Smith-form problem over the Laurent polynomial ring ℤ[q,q⁻¹]. For every a,b ≥ 2, the q-Carlitz matrix C(a,b,2;q) is proved to be unimodularly equivalent over ℤ[q,q⁻¹] to an explicit two-factor Smith form...

Prime-Pair Incidence Matrices: Almost-All Higher Prime Correlations and Pascal Transforms

This paper studies the binary prime-pair incidence matrix defined by (Aₙ)ₘ,ᵣ = 1ℙ(m − r)1ℙ(m + r), whose row index records the midpoint of a prime pair and whose column index records its half-gap. Rows therefore encode unequal Goldbach representations, while columns count prime p...

Roots in the General Riordan Group via Cyclic Substitution Norms

This paper gives a complete decision procedure for n-th roots in the general Riordan group over algebraically closed fields of characteristic 0, resolving Open Problem 3 of Calero-Sanz and Prieto-Martínez. The central mechanism is a cyclic-norm theorem for semidirect products (1...

Pascal Translation and Hasse Divided-Power Modules in Mixed Characteristic

This paper develops the characteristic-free algebraic theory of the generalized Pascal translation matrices Lₙ(a) and identifies their Hasse action with the regular module of a finite divided-power algebra. The principal result converts invariant-submodule questions for Pascal tr...

Universal Bell–Riordan Diagonal Equivalence and Structured Pascal–Stirling Matrices

This paper establishes a universal diagonal-equivalence theorem for shifted exponential Bell–Riordan blocks. For an arbitrary invertible formal substitution φ(x) = λx + ⋯ with unit linear coefficient λ, every associated shifted Bell block is proved to be unimodularly equivalent t...

Hankel Matrices of Linear Recurrences: Unimodular Congruence and Pell Applications

This paper develops a general unimodular reduction theory for Hankel matrices generated by scalar linear recurrences. Its principal theorem shows that every sufficiently large square Hankel section arising from a recurrence of order d is unimodularly congruent, over an arbitrary...

Pascal Translation and Hasse Divided-Power Modules in Mixed Characteristic

This paper develops the characteristic-free algebraic theory of the generalized Pascal translation matrices Lₙ(a) and identifies their Hasse action with the regular module of a finite divided-power algebra. The principal result converts invariant-submodule questions for Pascal tr...

Universal Bell–Riordan Diagonal Equivalence and Structured Pascal–Stirling Matrices

This paper establishes a universal diagonal-equivalence theorem for shifted exponential Bell–Riordan blocks. For an arbitrary invertible formal substitution φ(x) = λx + ⋯ with unit linear coefficient λ, every associated shifted Bell block is proved to be unimodularly equivalent t...

Truncated Toeplitz Unit Groups and Exact Reciprocal-Support Semigroups

This paper develops the algebraic theory of weighted lower-triangular Toeplitz matrices as finite truncated-convolution operators and determines exactly which coefficients can occur in their reciprocals. The principal result is a reciprocal-support theorem valid in characteristic...