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

TL;DR

This paper introduces a unified elliptic extension of Clausen functions, revealing deep structural parallels among circular, elliptic, and hyperbolic regimes, and clarifies their degeneration limits and boundary constants.

Contribution

It develops a new elliptic extension of Clausen functions based on Jacobi theta functions, establishing a clear structural framework and boundary constant organization across regimes.

Findings

Unified elliptic Clausen functions satisfy classical integral recursion.

Boundary constants form modular families linked to the elliptic kernel.

Degeneration limits between regimes are made transparent.

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.