Exact Analytical Regularization of Hyperbolic Asymptotic Divergences via the Binomial Series

The emergence of hyperbolic singularities of the 1/r type in gravitational potentials and fluid dynamics requires spatial softening techniques to prevent mathematical collapse at the origin. Traditionally, this splice is performed by approximating polynomials through the numerical resolution of systems of equations. In this methodological note, we demonstrate that the complete family of regularization polynomials guaranteeing a Ck class splice (continuity up to the k-th derivative) possesses a closed and exact analytical solution. This solution emerges naturally from the Newton binomial expansion and the combinatorial numbers of Pascal's Triangle, collapsing into the identity Pk(u)=[1−(1−u2/d2)k+1]/u. The physical implications of this transition are analyzed, demonstrating that the regularized core transforms the asymptotic sink into a finite-density harmonic oscillator, enabling the theoretical viability of acoustic waves in continuous media.