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

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.

💡 Research Summary

The paper presents a unified elliptic extension of the classical Clausen functions, preserving the integral recursion that characterizes the CL‑type family while isolating all regime‑dependent differences into boundary constants. Classical Clausen functions Clₙ(x) are defined by Fourier‑type series (sin k x/kⁿ for even n, cos k x/kⁿ for odd n) and satisfy the simple differential recursion d/dx Clₙ₊₁(x)=Clₙ(x). Their boundary constants arise from the primitive –log 2 sin(x/2), which links higher‑order constants to odd zeta values ζ(2m+1).

The author replaces the logarithmic sine kernel with the logarithm of the odd Jacobi theta function, defining the elliptic kernel Kₑₗₗ(x;τ)=log ϑ₁(x|τ). Because ϑ₁(x|τ) degenerates to 2 sin(πx) as τ→i∞ and to sinh(πx) after an S‑modular transformation (τ→i0⁺), Kₑₗₗ simultaneously encodes the circular, elliptic, and hyperbolic regimes.

Elliptic Clausen functions EClₙ(x;τ) are introduced recursively by ∂ₓEClₙ₊₁=EClₙ, with the initial function fixed by the primitive Kₑₗₗ. Consequently, the homogeneous part of the recursion is identical across all regimes; the only variation lies in the boundary constants B₂ₘ₊₁(τ)=ECl₂ₘ₊₁(0;τ). Expanding Kₑₗₗ near the origin yields Kₑₗₗ(x;τ)=−2 log