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

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.