Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Recursion Backbone for Circular and Elliptic Clausen Hierarchies

Ken Nagai

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.

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.
Paper Structure (37 sections, 2 theorems, 72 equations)

This paper contains 37 sections, 2 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. The Master Polylogarithmic Object and the SL-Type Family
  3. The Polylogarithmic Master Function
  4. CL-type and SL-type as Complementary Components
  5. Low-Order Examples
  6. Boundary Constants and Base-Point Normalization
  7. Elliptic CL/SL families from the normalized $\vartheta_1$ seed
  8. Normalized seed.
  9. Elliptic master family.
  10. Elliptic CL/SL components.
  11. Trigonometric degeneration: $\widetilde{\vartheta}_1 \to \sin(\pi z)$
  12. Bridge to the circular master via the trigonometric limit
  13. The second elliptic order
  14. Local expansion near $z=0$.
  15. First visualization of the elliptic SL-type component
  16. ...and 22 more sections

Key Result

Proposition 2.1

For all integers $n\ge1$,

Theorems & Definitions (10)

  • Proposition 2.1: Uniform Recursion Backbone
  • proof
  • Definition 1
  • Proposition 2.2: Parallel Recursion
  • proof
  • Remark 1: Boundary data
  • Remark 2: Branch choice for $\mathop{\mathrm{Arg}}\nolimits$
  • Remark 3: Trigonometric limit of the SL-seed
  • Remark 4: Backbone and generating viewpoints
  • Remark 5