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

Abstract

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 to a fixed diagonal determined entirely by binomial coefficients. The result extends from pure substitutions to the full invertible exponential Riordan class g(x)h(φ(x)), allowing an arbitrary unit constant term in the Riordan prefactor. As a specialization, the theorem resolves Callan’s unsigned Stirling-cycle diagonal-equivalence problem. The universal equivalence determines substantially more than the diagonal form. After arbitrary base change it gives the cokernel decomposition, all Fitting ideals, and residue-field ranks. For integral specializations it determines the complete p-primary Smith valuation data. These valuations are governed by base-p carries, producing an explicit two-state carry automaton and closed modular-rank formulas over 𝔽ₚ. The paper also develops the surrounding structured-matrix theory connecting Pascal, Stirling, and falling-factorial Vandermonde matrices. Results include exact LU and Cholesky factorizations, explicit inverses, minimal-polynomial and eigenspace information, the exact infinity-norm condition number κ∞(Lₚ) = 4ⁿ⁻¹, spectral bounds for the symmetric Pascal matrix, exact monomial–falling-factorial basis transformations, and Vandermonde–Stirling evaluation factorizations. These identities yield O(n²) construction and triangular-solve algorithms, finite-precision error bounds, Newton interpolation formulas, finite-difference identities, and combinatorial convolution formulas. The Bell–Riordan diagonal-equivalence theorem is the principal result; the Pascal–Stirling theory supplies its structural setting, concrete specializations, and computational consequences.