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Functional Equations for Generalized Collatz Dynamics Using Integral Representations and Residue Calculus

Christos N. Efrem

TL;DR

The paper addresses generalized Collatz-type dynamics by developing a unified complex-analytic framework to derive recursive relations and functional equations for their generating functions. It extends Egoryczhev-inspired contour-integral techniques to both one- and multi-dimensional settings, yielding explicit recurrences for $f_k(z)$ and the bivariate (or vector-valued) generating functions $F(z,w)$ and $oldsymbol{F}(oldsymbol{z},w)$. The main contributions are Theorems 1 and 2, which generalize Berg–Meinardus-type recursions to several complex variables and provide a structured pathway for residue-based evaluations in higher dimensions. This work broadens the toolkit for analyzing Collatz-type systems, offering a principled approach to derive functional equations rather than proving or disproving the conjecture itself, and it lays the groundwork for future multidimensional analyses and potential asymptotic insights.

Abstract

This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex analysis, including contour-integral representations, residue calculus, and Cauchy's integral formulas. The basic approach is inspired by Egorychev's method, which has been used so far to establish combinatorial identities. Moreover, this work generalizes existing results and extends the methodology to multiple dimensions using several complex variables.

Functional Equations for Generalized Collatz Dynamics Using Integral Representations and Residue Calculus

TL;DR

The paper addresses generalized Collatz-type dynamics by developing a unified complex-analytic framework to derive recursive relations and functional equations for their generating functions. It extends Egoryczhev-inspired contour-integral techniques to both one- and multi-dimensional settings, yielding explicit recurrences for and the bivariate (or vector-valued) generating functions and . The main contributions are Theorems 1 and 2, which generalize Berg–Meinardus-type recursions to several complex variables and provide a structured pathway for residue-based evaluations in higher dimensions. This work broadens the toolkit for analyzing Collatz-type systems, offering a principled approach to derive functional equations rather than proving or disproving the conjecture itself, and it lays the groundwork for future multidimensional analyses and potential asymptotic insights.

Abstract

This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex analysis, including contour-integral representations, residue calculus, and Cauchy's integral formulas. The basic approach is inspired by Egorychev's method, which has been used so far to establish combinatorial identities. Moreover, this work generalizes existing results and extends the methodology to multiple dimensions using several complex variables.

Paper Structure

This paper contains 27 sections, 20 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Closely Related Work
  3. Overview of Main Contributions
  4. Outline
  5. Notation
  6. Main Results
  7. One-Dimensional Arithmetic Dynamics
  8. Multi-Dimensional Arithmetic Dynamics
  9. Preliminaries: Convergence of Multiple Infinite Series
  10. Multiple Series of Complex Numbers
  11. Multiple Series of Complex-Valued Functions
  12. Power Series in Several Complex Variables
  13. Useful Lemmas
  14. Identities for Sums of Iverson's Brackets
  15. Integral Representations of the Kronecker Deltas
  16. ...and 12 more sections

Key Result

Proposition 2.1

Let $\rho_z, \rho_w \in \mathbb{R}_+$ such that $\rho_z < 1$ and $\rho_w < R_w$, where $R_w \coloneqq \min \{ {\left\lVert \mathbf{a} \right\rVert}_{\max}^{-1},1 \} \newline> 0$.Note that ${\left\lVert \mathbf{a} \right\rVert}_{\max} \neq 0$ (equivalently, ${\left\lVert \mathbf{a} \right\rVert}_{\ma

Theorems & Definitions (49)

  • Definition 2.1: Generalized Collatz dynamics
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • Example 2.1
  • Example 2.2
  • ...and 39 more