Generalized Cotangent Series and Links Zeta and Theta Functions

TL;DR

This paper introduces a generalized cotangent series that connects lattice sums, zeta functions, and theta functions, revealing new identities and structural relationships within modular forms and special functions.

Contribution

It extends classical cotangent expansions to higher-order lattice sums, deriving new closed-form representations and integral identities linking zeta and theta functions.

Findings

Derived new series representations for zeta values

Established integral identities connecting lattice sums and theta functions

Uncovered recursive patterns in generalized cotangent series

Abstract

This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine trigonometric and hyperbolic structures, uncover recursive patterns, and establish integral connections to generalized Jacobi theta functions. The analysis reveals deep structural relationships between lattice summations, values of the Riemann zeta function, and modular kernels of theta type. Using techniques such as contour integration, Mellin transforms, and factorization identities, we obtain new expressions for even zeta values and formulate integral identities linking these series to generalized theta functions. These results unify classical expansions of trigonometric and hyperbolic cotangent functions within a broader analytic framework, offering new insights into modular forms.