Exact Closed-Form Solutions for Trigonometric Integral Series via Kyungu Formulas

Description (English): This work introduces a family of exact analytic identities, designated as the Kyungu Formulas, which provide closed-form summation of trigonometric integral series: \[\sum_{n=1}^{\infty} \frac{\mathrm{Ci}(ns)}{n^{2k}} \quad \text{and} \quad \sum_{n=1}^{\infty} \frac{\mathrm{Si}(ns)}{n^{2k+1}}, \quad s>0, \, k \in \mathbb{N}^*\] These formulas constitute a universal analytic continuation over the positive real continuum and reveal an invariant structure combining Bernoulli polynomials, logarithmic phase terms, and jump operators. They provide rigorous expressions for oscillatory infinite series and extend the evaluation beyond the fundamental cycle 2\pi, offering exact results for arguments in all real domains. The Kyungu Formulas leverage the hypersummational operator, a tool introduced by the author in prior works, which generalizes classical summation techniques such as Euler–Maclaurin and Abel–Plana. By exploiting the intrinsic periodicity of Bernoulli kernels, the method transforms infinite series into finite combinations of algebraic constants and polynomial integrals, ensuring phase continuity and structural invariance independent of the order k. The work includes representative examples for various arguments and orders, demonstrating the applicability of the method for both Ci and Si series. The formulas also highlight connections with classical constants, including the Glaisher–Kinkelin constant, and provide a framework for deeper exploration in modern summatial analysis.