Biorthogonal eigenvectors of the Holte carry matrix and cascade-free enumeration

TL;DR

This paper provides a complete biorthogonal eigenvector system for the Holte carry matrix, explores its properties, and analyzes cascade-free enumeration, revealing conditions under which Chebyshev polynomial forms apply.

Contribution

It introduces explicit biorthogonal eigenvectors for the Holte matrix and characterizes cascade-free avoidance counts, extending understanding of carry process spectral properties.

Findings

Eigenvalues of the Holte matrix are simple and independent of N.

Explicit formulas for left and right eigenvectors are derived.

Chebyshev polynomial form holds for k=3 but not for k≥4.

Abstract

For $k$-summand base-$N$ addition, the carry process is a Markov chain on $\{0,\ldots,k-1\}$ whose transition matrix--the Holte matrix $T$--has eigenvalues $\{N^{-j}\}_{j=0}^{k-1}$, all simple and independent of $N$. We give the complete biorthogonal eigenvector system. The left eigenvectors factor as $\sum_i u_j[i] x^i = c_{k,j} (x-1)^j A_{k-j}(x)$, where $c_{k,j} = |s(k,k-j)|/k!$ involves unsigned Stirling numbers and $A_n(x)$ is the Eulerian polynomial. The right eigenvectors satisfy $\sum_i \binom{k-1}{i} v_j[i] x^i = (1+x)^{k-1-j} Q_j(x)$, where the quotient polynomials $Q_j$ have palindrome symmetry $x^j Q_j(1/x) = (-1)^j Q_j(x)$ and converge to $(1-x)^j$ as $k \to \infty$; for $j \le 3$, we give explicit closed forms in terms of $k$. The cascade-free avoidance count satisfies $a(L) = (\sqrt{d})^L U_L(x)$ (Chebyshev polynomial of the second kind) whenever the restricted transfer matrix has dimension $d \le 2$; we prove this is sharp: for $k$-summand addition, Chebyshev form holds for $k = 3$ and fails for $k \ge 4$. The proof uses oscillatory matrix theory to establish non-vanishing of all spectral residues. The characteristic polynomial of the restricted transfer matrix is determined in closed form by a Stirling-weighted Lagrange interpolation at the Holte eigenvalues. Two systems with binary carry state spaces are shadow-equivalent if and only if they share the pair $(N, d)$. The general classification for $k$-state systems reduces to the characteristic polynomial of $T$.