Global Convergence of Function-Modulated Transcendental Power Iteration Systems on the Complex Plane and Their Duality Structures

This paper investigates a class of generalized transcendental power iteration systems: u_{n+1} = u_n^{1/R(s)} e^{-π u_n / R(s)^2}, s ∈ C \ {0,1} where R(s) is a meromorphic function, extending the system parameter from a constant to a function module. Building on previous works [1, 2, 3], we prove that for almost all initial values u_0 ∈ C \ {0}, the iteration converges to a unique fixed point L_s satisfying the generalized balance equation: ln L_s + π/(R(s)(R(s)-1)) L_s = 0 Using the Lambert W function [4], we obtain the closed-form solution: L_s = R(s)(R(s)-1)/π · W_0( π/(R(s)(R(s)-1)) ) where W_0 denotes the principal branch. Furthermore, we extend the core structure 1/(R(s)(R(s)-1)) to an arbitrary meromorphic function Q(s), obtaining a more general system: ln M_s + π Q(s) M_s = 0 with solution: M_s = 1/(π Q(s)) · W_0(π Q(s)) For both extensions, we establish complete triple-space duality relations: Form A (R(s) parametrization): U_s = R(s)(R(s)-1)/π · W_0, V_s = e^{W_0}, Z_s = R(s)/π · W_0, W_0 = W_0(π/(R(s)(R(s)-1))) Form B (Q(s) parametrization): U_s = 1/(π Q(s)) · W_0(π Q(s)), V_s = e^{W_0(π Q(s))}, Z_s = s/π · W_0(π Q(s)) Both forms satisfy the duality relations U_s · V_s = 1, U_s = e^{-π Z_s / s}, V_s = e^{π Z_s / s}. When R(s) = s or Q(s) = 1/(s(s-1)), the systems reduce to [3]. These results reveal a unified structure for function-modulated transcendental dynamical systems, providing new paradigms for the intersection of complex dynamics and special function theory [5, 6, 7, 8, 9].