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

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.