A Kyungu Summation Formula Based on Polygamma Transformations

This paper presents a new Kyungu summation formula derived through a local analytic transformation and the classical theory of polygamma functions. The proposed identity expresses finite sums exactly in terms of the derivatives of a transformed function evaluated at a single point, together with universal coefficients involving polygamma functions. Unlike the Euler–Maclaurin formula, the resulting representation contains no remainder term. A complete proof is provided, based on the Taylor expansion of the transformed function, the linearity of finite summation, and the classical representation of generalized harmonic numbers by polygamma functions. The paper also discusses the analytical significance of the formula, compares it with classical summation methods, and illustrates its scope through several examples. This work constitutes a rigorous proof of a previously introduced Kyungu summation formula and further enriches the theoretical framework of Summatial Analysis.