The LMₙ(x) Function: A Generalized Inverse of Iterated Exponential Functions

Abstract

This work introduces and develops the LMₙ function as an inverse framework associated with iterated exponential and power-type functions. The paper develops the definition and basic properties of LMₙ, with particular attention to its recursive structure, inverse relationships, real-valued branches, and special cases. The case LM₂ is studied in detail through the self-power function xˣ, including its monotonicity, critical point, real branches, and connection with the Lambert W function. Higher-order cases are then considered through the corresponding recursive functions. The work also investigates differential and integral properties, inverse-function derivatives, Taylor expansions and Lagrange remainder estimates, numerical approximation, asymptotic behaviour, and the relationship between LMₙ, tetration, super-roots, super-square-roots, and related constructions. The purpose of this work is to provide a systematic mathematical framework for studying inverse functions of iterated exponential structures and to identify directions for further analytical and computational investigation.