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Elliptic Clausen Functions and Degenerations Circular, Elliptic, and Hyperbolic Parallelism

Ken Nagai

TL;DR

The paper develops a unified elliptic extension of CL-type Clausen functions by defining elliptic Clausen functions $\operatorname{ECl}_n(x;\tau)$ through the same integral recursion as the classical Clausen functions, but with boundary data fixed by the elliptic logarithmic kernel $K_{\mathrm{ell}}(x;\tau)=\log\vartheta_1(x|\tau)$. It shows that the recursion structure is regime-independent, while the differences among circular, elliptic, and hyperbolic cases reside entirely in the boundary constants, which are built from logarithmic primitives $\log\sin$, $\log\vartheta_1$, and $log\sinh$. The circular and hyperbolic degeneration limits are made explicit: as $\tau\to i\infty$ one recovers the classical Clausen functions $\operatorname{Cl}_n$, and via modular $S$-transforms the hyperbolic regime emerges with $\vartheta_1(x|\tau)\to\sinh(\pi x)$, giving $\operatorname{HCl}_n$. The paper also details the modular structure of the odd boundary constants $\mathcal{B}_{2m+1}(\tau)$, their interpretation as moments of the elliptic kernel, and their reductions to zeta-values in degeneration limits, outlining potential extensions to SL-type Clausen functions and related master objects.

Abstract

We introduce a unified elliptic extension of CL-type Clausen functions based on logarithmic primitives of the Jacobi theta function. The resulting elliptic Clausen family satisfies the same integral recursion as the classical circular case, with all differences encoded in boundary constants determined by the underlying logarithmic kernel. This separation clarifies a strict parallelism between circular, elliptic, and hyperbolic regimes and makes their degeneration limits transparent. We further discuss the general structure of the odd boundary constants, which organize naturally into modular families associated with the elliptic kernel. Possible extensions to SL-type frameworks and related master objects are briefly outlined.

Elliptic Clausen Functions and Degenerations Circular, Elliptic, and Hyperbolic Parallelism

TL;DR

The paper develops a unified elliptic extension of CL-type Clausen functions by defining elliptic Clausen functions through the same integral recursion as the classical Clausen functions, but with boundary data fixed by the elliptic logarithmic kernel . It shows that the recursion structure is regime-independent, while the differences among circular, elliptic, and hyperbolic cases reside entirely in the boundary constants, which are built from logarithmic primitives , , and . The circular and hyperbolic degeneration limits are made explicit: as one recovers the classical Clausen functions , and via modular -transforms the hyperbolic regime emerges with , giving . The paper also details the modular structure of the odd boundary constants , their interpretation as moments of the elliptic kernel, and their reductions to zeta-values in degeneration limits, outlining potential extensions to SL-type Clausen functions and related master objects.

Abstract

We introduce a unified elliptic extension of CL-type Clausen functions based on logarithmic primitives of the Jacobi theta function. The resulting elliptic Clausen family satisfies the same integral recursion as the classical circular case, with all differences encoded in boundary constants determined by the underlying logarithmic kernel. This separation clarifies a strict parallelism between circular, elliptic, and hyperbolic regimes and makes their degeneration limits transparent. We further discuss the general structure of the odd boundary constants, which organize naturally into modular families associated with the elliptic kernel. Possible extensions to SL-type frameworks and related master objects are briefly outlined.
Paper Structure (16 sections, 2 theorems, 14 equations)

This paper contains 16 sections, 2 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Classical Clausen Functions and Integral Recursions
  3. Boundary constants in the circular case
  4. Motivation for elliptic extension
  5. Elliptic Kernel and Elliptic Clausen Functions
  6. Elliptic logarithmic kernel
  7. Definition of elliptic Clausen functions
  8. Degenerations and Boundary Constants
  9. Circular degeneration ($\tau\to i\infty$)
  10. Hyperbolic degeneration ($\tau\to i0^+$)
  11. Correspondence Table and Structural Overview
  12. Parallel correspondence
  13. Structural interpretation
  14. Remarks on the General Structure of $\mathcal{B}_{2m+1}(\tau)$
  15. General structure of the odd boundary constants
  16. ...and 1 more sections

Key Result

Proposition 1

Across circular, elliptic, and hyperbolic regimes, the integral recursion for CL-type Clausen functions is identical. All differences are encoded in the boundary constants, which are given by

Theorems & Definitions (4)

  • Proposition 1
  • Remark 1
  • Corollary 1
  • Remark 2