Functions and Analysis

Asymptotic Regularization of Poles and Closed-Form Generalization for Arithmetic Progressions in the Gamma Function

Authors: Julinho Jorge Luís
Comments: 25 Pages.

This paper proposes a novel analytical framework for the Gamma and Factorial functions, extending their consistent application beyond the domain of positive real numbers. Traditionally, these functions encounter singularities (poles) at non-positive integers, limiting their continuity. Through an asymptotic regularization method, this work demonstrates that the ratio of Gamma functions can yield finite and unique values at these critical points, effectively bypassing traditional meromorphic constraints.The core contribution is the derivation of a universal closed-form formula for the product of arithmetic progressions, valid across the entire real line without the need for manual domain adjustments. Furthermore, the concept of a "Rising Gamma Function" ($check{Gamma}$) is introduced as a dual operator. By establishing the zero point as an inversion axis, a functional symmetry is revealed, integrating the properties of the function into a complete and continuous structure. This approach provides new insights into analytical continuity and simplifies calculations in complex analysis and number theory.Keywords: Gamma Function, Factorial, Asymptotic Regularization, Arithmetic Progressions, Functional Symmetry, Analytical Continuity.
Category: Functions and Analysis