Hankel Matrices of Linear Recurrences: Unimodular Congruence and Pell Applications

Abstract

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 commutative ring, to its fixed leading d × d recurrence core together with a zero block. Consequently, the determinantal ideals, Fitting ideals, kernel and cokernel structure, nonzero Smith data over a PID, rank over a field, and nonzero real inertia of every such Hankel section reduce to invariants of a single finite core. The theory extends without loss to rectangular Hankel sections. Every m × n Hankel matrix generated by a scalar recurrence of order d, with m,n ≥ d, is unimodularly equivalent to its d × d core padded by zeros. Thus all determinantal ideals agree with those of the recurrence core up through order d, all higher determinantal ideals vanish, and the nonzero Smith invariant factors over a PID are exactly those of the core. A multivariate version is also established. For arrays satisfying independent scalar recurrences of orders d₁,…,d_q in their respective coordinates, every sufficiently large multilevel Hankel section reduces to a fixed core of size ∏ⱼ dⱼ. The complete determinantal, Smith, rank, and real-inertia data of arbitrarily large multilevel sections are thereby encoded by this finite recurrence core. This provides a tensorized recurrence-Hankel reduction principle for multidimensional structured arrays. The order-two theory is worked out explicitly. For the universal recurrence uₖ₊₂ = P uₖ₊₁ − Q uₖ, the paper derives the complete shifted 2 × 2 Hankel-minor identity, proves the vanishing of all higher minors, determines the recurrence kernel, and gives the resulting module reductions. When Q is a unit, the associated Hankel matrix is unimodularly equivalent to diag(1,1,0,…,0). Pell numbers provide the principal arithmetic specialization. For every shifted Pell-Hankel matrix Hₙ⁽ᵐ⁾, the Binet decomposition gives an exact rank-two outer-product representation and a complete recurrence kernel. Every 2 × 2 minor is obtained explicitly, and for n ≥ 2 the integer Smith form is determined. The rank is therefore exactly two over every field. The real inertia depends completely on the parity of the shift: even shifts have one positive and one negative eigenvalue, whereas odd shifts have two positive eigenvalues; all remaining eigenvalues vanish. In particular, the shifted Pell-Hankel matrix is positive semidefinite precisely for odd shifts. The characteristic polynomial and both nonzero eigenvalues are also obtained in closed form. The paper places these Hankel results within the finite-dimensional translation theory of weighted Pascal matrices. For Lₙ(a)ᵣ,𝑐 = aʳ⁻ᶜ C(r−1,c−1), the additive group law Lₙ(a)Lₙ(b) = Lₙ(a+b), inverse Lₙ(a)⁻¹ = Lₙ(−a), nilpotent infinitesimal generator, exponential representation, similarity structure, exact ∞-norm conditioning, row and column generating functions, and Gram-matrix formulas are developed explicitly. The specialization a = 2 connects the weighted Pascal translation structure directly with the Pell recurrence. Further results treat finite exponential sums, distinct-mode Hankel factorizations, block constructions, generating functions, Gram matrices, and exact finite-dimensional structured identities. Together these results establish recurrence order—not matrix dimension—as the fundamental algebraic complexity governing Hankel matrices generated by linear recurrences.