Z-Mahler Equations in Zeckendorf Numeration

• Z-Mahler equations are defined using the Zeckendorf numeration system, replacing radix powers with Fibonacci-based shifts.
• They provide a framework uniting automata theory, transcendence, and algorithmic methods through weighted Zeckendorf expansions.
• These equations reveal deep connections with spectral properties in quantum/random walks and extend classical rigidity results in arithmetic dynamics.

Z-Mahler equations constitute a generalization of classical Mahler functional equations, in which the combinatorial structure of the underlying numeration system—specifically, the Zeckendorf expansion via the Fibonacci sequence—replaces the conventional radix-$q$ powers. These equations provide a unifying framework for analyzing regularity and automata-theoretic properties of sequences indexed by nonstandard numeration schemes. The term “Z-Mahler equation” refers both to functional identities arising in quantum/random walk spectral correspondences and to difference equations over formal power series whose “shift” operator is governed by Zeckendorf numeration. The study of Z-Mahler equations encompasses automata theory, transcendence questions, algorithmic solution finding, and connections to Mahler measures in arithmetic dynamics.

1. Zeckendorf Numeration and Z-Mahler Operators

Zeckendorf numeration encodes each nonnegative integer $N$ uniquely as a sum of distinct non-consecutive Fibonacci numbers: $$N = \sum_{i=0}^k b_i F_i,\quad b_i\in\{0,1\},\, b_i b_{i+1}=0,$$ where $(F_n)_{n\ge0}$ is the Fibonacci sequence, $F_{-1}=1, F_0=0, F_{n+2}=F_{n+1}+F_n$. The canonical Zeckendorf expansion $(N)_Z$ is the word $b_k b_{k-1} \cdots b_0$ without two consecutive $1$'s.

The Z-Mahler operator $\Phi$ on formal power series acts according to the Zeckendorf shift: $$\varphi\left(\sum_i b_i F_i\right) = \sum_i b_i F_{i+1},$$ and extended linearly,

$$\Phi\left(\sum_{n\ge0} f_n x^n\right) = \sum_{n\ge0} f_n x^{\varphi(n)}.$$

A Z-Mahler equation of exponent $d$ and height $h$ is

$P(x, y) = \sum_{i=0}^{d} A_i(x) \Phi^i(y) = 0,$

where $A_i(x) \in R[x]$ for a commutative ring $R$ (Carton et al., 3 May 2024).

2. Automata-Theoretic Regularity: Z-Regular Sequences

A sequence $(a_N)_{N\ge0}$ is Z-regular if there exists a weighted Zeckendorf automaton $\mathcal{A} = (S,B,\Delta,I,F)$, assigning weights to the canonical Zeckendorf expansion $(N)_Z$, such that

$a_N = \mathrm{weight}_\mathcal{A}((N)_Z).$

Z-regularity generalizes the classical $q$-regular/automatic sequences by utilizing the state transitions corresponding to non-consecutive Fibonacci digits. Kernel methods and automaton constructions yield the equivalence:

• If $(a_N)$ is Z-regular, its generating function $f(x)=\sum a_N x^N$ satisfies a Z-Mahler equation.
• If $f(x)$ is the unique solution to an isolating Z-Mahler equation ($A_0=1$), then $(f_N)$ is Z-regular (Carton et al., 3 May 2024).

Explicit automata constructions for isolating Z-Mahler equations provide $O(h^2 d)$ state bounds.

3. Equation Structure, Solution Spaces, and Algorithmic Computation

Z-Mahler equations typically appear in the form

$f(x) = \sum_{i=1}^d A_i(x) \Phi^i(f(x)),$

with solution spaces governed by the structure of the operator $\Phi$ and the recurrence relations implicit in Zeckendorf numeration. The injectivity and near-additivity of $\varphi$ ($$\varphi(m) + \varphi(n) - \varphi(m+n)\in\{-1,0,1\}$$) leads to intricate recursive behaviors.

For solutions $f(x)$ in $R[[x]]$, isolating equations guarantee a (unique) Z-regular sequence, but non-isolating cases can exhibit growth exceeding all polynomial bounds (e.g., stretched exponential), hence violate Z-regularity criteria. The prototypical non-isolating example is $(1-x)f(x) = \Phi(f(x))$ (Carton et al., 3 May 2024).

Algorithmic solution to general Mahler equations—including Z-Mahler equations viewed as $p$-Mahler equations with $p$ arbitrary—proceeds via reductions:

• Newton polygon analysis to bound solution valuations and degrees,
• Recursion on initial terms via linear algebra,
• Extraction of rational, polynomial, or Puiseux series solutions by constructing companion systems and applying cyclic-vector methods,
• Assembly of Hahn-series/Puiseux/Hahn/constant basis representations for full solution sets (Faverjon et al., 24 Nov 2025, Chyzak et al., 2016).

Complexity for constructing truncated solution matrices, validating regular singularities (via Newton polygons, and truncated Frobenius series), and solving companion systems is polynomial in operator order, degree, and truncation size (Faverjon et al., 24 Feb 2025).

4. Relation to Classical Mahler Equations and Generalizations

Classical Mahler equations are functional equations of the form

$$P_0(z) F(z) + P_1(z) F(z^k) + \cdots + P_d(z) F(z^{k^d}) = 0,$$

with $k\ge2$ and $P_i(z)\in K[z]$ not all zero. Their extension to Z-Mahler equations rests on replacing radix-based power shifts $z \to z^{k^i}$ with shifts defined by Zeckendorf numeration.

Key rigidity results include:

• A formal power series $F(z)$ satisfying both $k$-Mahler and $\ell$-Mahler equations (with $k,\ell$ multiplicatively independent) is rational (Adamczewski et al., 2013).
• Z-Mahler equations in the Zeckendorf context are conjectured to display analogous rigidity—for example, if a series is both Z-Mahler and $q$-Mahler (for multiplicatively independent $q$), must it be rational? This remains open (Carton et al., 3 May 2024).

The linkage between Mahler equations and regular/automatic sequences is reflected in automata-theoretic characterizations: Mahler equations with integer coefficients generate characteristic series for automatic sets, and solution methods leverage automaton structures for efficient resolution (Chyzak et al., 2016).

5. Z-Mahler Identities in Random and Quantum Walks

The Mahler/Zeta Correspondence establishes explicit identities—termed “Z-Mahler equations”—between Mahler measures of specific Laurent polynomials and zeta-functions encoding spectral properties of discrete walks. For a $d$-dimensional random or quantum walk, such zeta functions are expressible as: $$L_{QW}(u) = \log(\mathcal{C}(u)) + m(P(X_1,\ldots,X_d;u)),$$ where $m(P)$ is the Mahler measure of a Laurent polynomial, and $\mathcal{C}(u)$ is a walk-dependent normalization factor (Komatsu et al., 2022). Detailed forms are given for various walk types and dimensions, e.g.,

$$L_{RW}(u) = \log(-\tfrac{u}{2}) + m(X + X^{-1} - 2u)$$

for the one-dimensional symmetric random walk.

This correspondence bridges arithmetic, spectral, and combinatorial properties, yielding connections to $L$-values and hypergeometric series via the Mahler measure.

6. Analytic and Combinatorial Aspects of Z-Mahler Functions

Mahler equations often admit distinct analytic incarnations: for instance, Wadim Zudilin’s work on the equation $U(q) = 1 + (q-1) U(q^2)$ provides both analytic power series solutions (convergent within the unit disk) and oscillatory, finitely supported series defined solely on roots of unity (Zudilin, 20 Mar 2024). The celebrated “strange identity” equates derivatives of both solutions at all orders precisely at each root of unity.

Such phenomenon suggests a hidden unity among the analytic, combinatorial, and automaton-theoretic facets of Mahler-type functions, with deep implications for transcendence and algebraic independence.

7. Open Problems and Research Directions

Current lines of inquiry include:

• Characterizing Z-regularity for geometric series $\sum \alpha^n x^n$ (in the radix-$q$ case, $\alpha$ must be a root of unity).
• Extending Z-Mahler theory to Pisot recurrences and other canonical numeration systems.
• Establishing Cobham-type theorems for Z-Mahler and $q$-Mahler equations with independent $q$ (rationality constraints).
• Exploring multidimensional generalizations (e.g., Mahler measures arising in spectral correspondences for higher-dimensional quantum walks).
• Algorithmic efficiency enhancements for solution computation, especially in Puiseux or Hahn series rings relevant in regular singularity analysis (Faverjon et al., 24 Feb 2025, Faverjon et al., 24 Nov 2025).

The research on Z-Mahler equations highlights the interaction between functional equations, automata, transcendence theory, and discrete dynamical systems, underscoring the rich interface of modern arithmetic combinatorics, algorithm design, and mathematical physics.