Holomorphic solution spaces of the functional equations satisfied by generating functions of generalized Collatz dynamics

For a wide class of generalized Collatz maps — piecewise-affine maps $t(n)=a_r n+b_r$ applied when $n\equiv r\pmod m$ , with arbitrary modulus $m$ and admissible rational coefficient vectors, and their multi-dimensional iterated analogues — Efrem (2025) derived, via Egorychev-style contour-integral and residue calculus, functional recurrences and bivariate/multivariate functional equations satisfied by the ordinary generating functions of the iterates $t_k(n)$ . The paper, which explicitly generalizes the Berg–Meinardus functional-equation approach to the classical $3n+1$ problem, deliberately does not analyze the equations it produces. The present statement formulates the problem the paper leaves open: (i) characterize structurally and analytically the solution spaces of these homogeneous and inhomogeneous functional equations (in one and several complex variables) on their natural domains; (ii) determine for which generalized Collatz dynamics such a characterization is equivalent to, or otherwise decides, orbit-termination and other global dynamical properties — generalizing Berg–Meinardus's theorem that, for the classical map, the corresponding solution-space statement is equivalent to the Collatz conjecture; and (iii) obtain explicit closed-form (non-inductive) descriptions of the multivariate partial-fraction data entering the multi-dimensional equations. The problem is open; its classical instance is provably as hard as the Collatz conjecture.

Background

The Collatz map (or $3n+1$ map) acts on the nonnegative integers by sending even $n$ to $n/2$ and odd $n$ to $3n+1$ ; since $3n+1$ is even for odd $n$ , one usually studies the shortened Collatz map $\psi$ sending even $n$ to $n/2$ and odd $n$ to $(3n+1)/2$ . The Collatz conjecture asserts that every $n\geq 1$ eventually reaches $1$ under iteration; it remains open despite extensive surveys, probabilistic results such as Tao's theorem that almost all orbits (in logarithmic density) attain almost bounded values, and massive numerical verification.

An analytic thread on the conjecture, initiated by Berg and Meinardus (1994, 1995), works with generating functions: for each iterate $\psi_k$ one forms the power series $h_k(z)=\sum_{n\geq 0}\psi_k(n)z^n$ , which converges on the open unit disk $|z|<1$ (a power series converging on a disk defines a holomorphic, i.e. complex-differentiable, function there), and the bivariate series $H(z,w)=\sum_{k,n}\psi_k(n)z^n w^k$ . They showed these functions satisfy exact functional equations, e.g. $h_k(z)=h_{k-1}(z^{2})+\frac{1}{3z^{1/3}}\sum_{l=0}^{2}\zeta^{l}h_{k-1}(z^{2/3}\zeta^{l})$ with $\zeta=e^{2\pi i/3}$ , and — the anchor result of this thread — proved that the Collatz conjecture is equivalent to the statement that the only solutions of the associated homogeneous equation (obtained from this recurrence by replacing $h_k$ and $h_{k-1}$ with a single function $h$ ) holomorphic on $|z|<1$ are $h(z)=c_0+c_1 z/(1-z)$ . Thus, in this framework, characterizing a solution space is deciding the conjecture. Berg and Opfer (2013) extended the analysis to negative start values ( $3n-1$ ).

Efrem (2025, arXiv:2510.06736) generalized this program. His Definition 2.1 introduces the $(m,\mathbf{a},\mathbf{b})$ -Collatz map $t:\mathbb{N}_0\to\mathbb{N}_0$ , $t(n)=a_r n+b_r$ if $n\equiv r\pmod m$ , where the rational parameters are constrained exactly so that $t$ maps nonnegative integers to nonnegative integers ( $\lambda_r:=a_r m\in\mathbb{N}$ and $\mu_r:=a_r r+b_r\in\mathbb{N}_0$ for each residue $r$ ); the classical shortened map is the instance $m=2$ , $\mathbf{a}=[1/2,3/2]$ , $\mathbf{b}=[0,1/2]$ . A multi-dimensional version (Definition 2.2) replaces $n$ by a vector iterated through a matrix recursion. Using the Egorychev method — a technique from combinatorial analysis in which coefficients are represented by contour integrals and extracted by residue calculus (computing integrals via the singularities they enclose) — he proved (Theorems 2.1, 2.2) that the generating functions $f_k(z)=\sum_n t_k(n)z^n$ and $F(z,w)=\sum_{k,n}t_k(n)z^n w^k$ satisfy a contour-integral recursion, a functional recurrence, and a bivariate functional equation of the form

$F(z,w)=f_0(z)+w\sum_{r=0}^{m-1}z^{r}\Big(\sum_{l}\frac{\sigma_{r,l}(z)}{l!}\,F^{(l)}_*(0,w)+\sum_{l}\tau_{r,l}(z)\,F(\phi_{r,l}(z),w)\Big)$

where the coefficients $\sigma_{r,l},\tau_{r,l}$ come from an explicit partial-fraction decomposition (the unique decomposition of a rational function into simpler fractions) and $\phi_{r,l}(z)=z^{m/\lambda_r}e^{2\pi i l/\lambda_r}$ are root maps; a several-variable analogue involves index sets $\mathcal{L}(\mathbf{r})$ and coefficient functions $\eta_{\mathbf{r},\boldsymbol{\ell},\nu}$ , $\boldsymbol{\varphi}_{\mathbf{r},\boldsymbol{\ell},\nu}$ that the paper describes only through an inductive construction on the dimension. The paper states plainly that its purpose is to generalize Berg–Meinardus–Opfer rather than to prove or disprove the Collatz conjecture, and it nowhere analyzes the equations it derives: the solution spaces are not characterized, no dynamical consequence is drawn, and no closed form for the multi-dimensional data is given. A separate line of work (Siegel, 2019–2026) develops different functional equations for Collatz-type 'Hydra maps' via complex and $p$ -adic analysis (analysis over the field of rational numbers completed with a $p$ -based notion of distance), but its objects are not these generating functions. For context on the outer boundary: Conway's iteration-universality results show that the termination question for the class of all generalized Collatz-type functions is undecidable in general, so any solution-space-to-dynamics correspondence must be established per admissible subfamily rather than uniformly.

Problem Statement

For the functional equations derived by Efrem (2025) for generalized Collatz dynamics — the functional recurrence for $f_k(z)$ , the one-dimensional bivariate equation of Theorem 2.1(S1.c) for $(m,\mathbf{a},\mathbf{b})$ -Collatz maps and its multivariate analogue in Theorem 2.2, together with their homogeneous associates (for the bivariate equation, the equation obtained by dropping the additive term $f_0(z)$ ; for the functional recurrence, the stationary homogeneous equation obtained by identifying $f_k$ and $f_{k-1}$ with a single function, as the source's equation (6) does in the classical case, and to which the Berg–Meinardus equivalence attaches) — establish an analytic theory of their solutions, with the following jointly linked objectives serving one goal (turning the derived equations into usable information about the dynamics):

Solution-space characterization. Determine the structure of the set of holomorphic solutions on the natural domains (the unit disk or punctured disk in one variable, the corresponding polydisks (products of disks) in several variables): for the inhomogeneous equation, describe all solutions and the extent to which the true generating function is distinguished among them; for the homogeneous equation, determine all solutions or sharp structural constraints on them. A solution-space characterization is likewise sought for explicit admissible subfamilies of parameters $(m,\mathbf{a},\mathbf{b})$ beyond the classical instance covered by Berg–Meinardus.
Dynamical content. For which admissible parameter families is a solution-space characterization equivalent to, or implies, a global dynamical statement about the orbits of the underlying map (orbit termination/reachability of cycles, in the spirit of Berg–Meinardus's equivalence between the classical solution-space statement and the Collatz conjecture)? Establish such equivalences, or prove separation results showing the functional equations alone do not encode the dynamics.
Explicit coefficient data. Provide a closed-form, non-inductive description of the multivariate partial-fraction data $\mathcal{L}(\mathbf{r})$ , $\eta_{\mathbf{r},\boldsymbol{\ell},\nu}(\mathbf{z})$ , $\boldsymbol{\varphi}_{\mathbf{r},\boldsymbol{\ell},\nu}(\mathbf{z})$ of Theorem 2.2 and of their dependence on the defining matrix parameters, replacing the inductive construction of Section 8.1 of the source paper, which makes these data available only recursively (footnote 7 to Theorem 2.2); such a description makes the multi-dimensional equations explicitly usable in components 1–2.

The classical shortened $3n+1$ map is the instance $(m,\mathbf{a},\mathbf{b})=(2,[1/2,3/2],[0,1/2])$ ; by Berg–Meinardus (1994, Theorem 5), component 2 for the classical instance of the recurrence-derived homogeneous equation is equivalent to the Collatz conjecture, and the problem is stated at the source's full generality without any restriction imposed for convenience.

The three components are stated at the source's full admissible generality; at that generality the general statements of components 1 and 2 are Collatz-hard and remain open, while component 3 is the finite constructive result that makes the multi-dimensional equations explicitly usable. Three strictly weaker sub-claims stand independently — the explicit closed form of the multivariate partial-fraction data $\mathcal{L}(\mathbf{r})$ , $\eta_{\mathbf{r},\boldsymbol{\ell},\nu}(\mathbf{z})$ , $\boldsymbol{\varphi}_{\mathbf{r},\boldsymbol{\ell},\nu}(\mathbf{z})$ (component 3); a solution-space characterization for any one explicit non-classical admissible family $(m,\mathbf{a},\mathbf{b})$ (component 1); and a dynamical equivalence or separation theorem for any one admissible family (component 2) — and none of them by itself implies the general statements of components 1 and 2.

References

Christos N. Efrem (2025), "Functional Equations for Generalized Collatz Dynamics Using Integral Representations and Residue Calculus", arXiv:2510.06736, DOI 10.48550/arXiv.2510.06736, https://arxiv.org/abs/2510.06736
Lothar Berg and Günter Meinardus (1994), "Functional equations connected with the Collatz problem", Results in Mathematics 25, No. 1-2, 1-12, DOI 10.1007/bf03323136, https://doi.org/10.1007/bf03323136, zbMATH 0810.11013
Lothar Berg and Günter Meinardus (1995), "The 3n+1 Collatz problem and functional equations", Rostocker Mathematisches Kolloquium 48, 11-18, zbMATH 0861.11008, https://zbmath.org/0861.11008
Lothar Berg and Gerhard Opfer (2013), "An Analytic Approach to the Collatz 3n+1 Problem for Negative Start Values", Computational Methods and Function Theory 13, 225-236, DOI 10.1007/s40315-013-0017-z, https://doi.org/10.1007/s40315-013-0017-z
Maxwell Charles Siegel (2019), "Conservation of Singularities in Functional Equations Associated to Collatz-Type Dynamical Systems; or, Dreamcatchers for Hydra Maps", arXiv:1909.09733, https://arxiv.org/abs/1909.09733
Maxwell C. Siegel (2026), "The Hydra Map and Numen Formalisms for Collatz-Type Problems", arXiv:2601.17030, https://arxiv.org/abs/2601.17030
Terence Tao (2022), "Almost all orbits of the Collatz map attain almost bounded values", Forum of Mathematics, Pi 10, e12, DOI 10.1017/fmp.2022.8, https://doi.org/10.1017/fmp.2022.8
J. H. Conway (2010), "Unpredictable iterations", in Jeffrey C. Lagarias (ed.), The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, ISBN 978-0-8218-4940-8, 219-223, zbMATH 1216.11109, https://zbmath.org/1216.11109 (original 1972 Boulder proceedings version: zbMATH 0337.10041)