Roots in the General Riordan Group via Cyclic Substitution Norms

Abstract

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 + 𝔪) ⋊ Aut(A) over complete filtered commutative ℚ-algebras. Once an n-th root of the substitution component has been chosen, the remaining multiplicative root equation is linearized by the formal logarithm and reduced to the image problem (I + τ + ⋯ + τⁿ⁻¹)A = L. This reduction converts Riordan root extraction into an explicit problem for a cyclic substitution operator. The resulting image problem is solved in all possible dynamical regimes. When the multiplier is not a root of unity, the cyclic norm is surjective and root extraction is automatic after the substitution root is fixed. For genuinely resonant substitutions of infinite order, existence is characterized by a finite and explicitly computable list of coefficient obstructions concentrated in the lowest resonant sector. For finite-order substitutions, the entire obstruction collapses to a single invariant divisibility condition. Together these cases exhaust the general Riordan root problem and produce an effective root-existence criterion. The cyclic-norm theorem is formulated at the level of complete filtered algebras, separating the underlying semidirect-product mechanism from its Riordan specialization. This exposes root extraction as a linear image problem after logarithmic passage and provides a framework applicable to other substitution-driven algebraic structures. The paper also develops a complementary calculus of generating functions and truncated convolution algebras arising from structured matrix families. Lower-triangular Toeplitz multiplication is identified with Cauchy convolution of first columns, while row-sum generating functions satisfy R_AB(x) = (1 − x)R_A(x)R_B(x). Generalized Pascal matrices Bₐ satisfy the additive semigroup law BₐB_b = Bₐ₊b. Compatible algebra homomorphisms of truncated power-series systems are classified as formal substitutions; compatible derivations are exactly operators h(z)d/dz with h(z) ∈ zR[[z]]; and compatible regular-module maps are precisely formal-series multipliers. Further applications cover Pascal, Stirling, Pell, Fibonacci, central-binomial, Catalan, Vandermonde, and related structured families. Their ordinary and exponential generating functions are analyzed together with truncation behavior, convolution identities, Dirichlet-series obstructions, and asymptotic regimes determined by polynomial termination, dominant simple poles, and square-root branch points.