A Recursion Backbone for Circular and Elliptic Clausen Hierarchies

TL;DR

This paper develops a recursive framework to extend Clausen-type functions into elliptic analogues, unifying their structure and providing a new perspective on their interrelations.

Contribution

It introduces an elliptic extension of Clausen functions using a recursive approach, connecting circular and elliptic cases through a unified framework.

Findings

Constructed elliptic Clausen functions from polylogarithmic master functions.

Established a structural correspondence between circular and elliptic functions.

Presented a generating deformation that unifies the recursion into a single analytic object.

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.