Summatial Analysis: Exact Closed-Form Evaluation of Bessel-Type Series via the Hyper-summatial

🇫🇷🇬🇧 Description: This work presents an advanced development of Summatial Analysis, a mathematical framework for the exact evaluation of structured infinite series using the Hyper-summatial operator. Building upon the foundational theory, this version introduces a structural filtering principle based on Bernoulli-type mechanisms, leading to the systematic annihilation of higher-order odd polynomial components. This filtering reveals the intrinsic analytic content of the series and significantly simplifies their evaluation. The core contribution of this work is the derivation of exact closed-form summation formulas for Bessel-type series. These results establish a direct and explicit connection between discrete summations and continuous integral representations, without requiring traditional regularization or contour integration techniques. The Hyper-summatial operator acts on auxiliary functions derived from inverse Laplace transforms, allowing the transformation of complex series into analytically tractable forms. This approach leads to precise corrective terms involving exponential structures of the form (e^2 a-1) ^-1, which naturally emerge from the underlying analytic framework. The paper includes both English and French versions, ensuring broader accessibility and dissemination of the theory. Summatial Analysis provides a unified structural approach to summation theory, bridging discrete mathematics, integral transforms, special functions, and complex analysis. This framework opens new perspectives for the exact treatment of series involving Bessel functions and suggests further extensions to more general classes of special functions.