A Recursion Backbone for Circular and Elliptic Clausen Hierarchies

Abstract

We introduce an elliptic extension of Clausen-type functions based on a unified recursive framework. Starting from the polylogarithmic master function, we construct a pair of circular functions whose real and imaginary parts correspond to the classical Clausen-type structures. Replacing the trigonometric seed with a Jacobi theta function yields an elliptic deformation that preserves the same recursive backbone. The circular limit recovers the original functions, establishing a structural correspondence between the circular and elliptic settings. Furthermore, we introduce a generating deformation that organizes the recursion into a single analytic object. This viewpoint suggests a unified framework for Clausen-type functions and their elliptic analogues.