Binomial Law for Bessel Function Series

🇬🇧 This work introduces a binomial structural law governing infinite series weighted by Bessel functions. By combining summation analysis, Bernoulli polynomial periodization, and a finite jump-operator framework, the study establishes exact closed-form expressions for series of the form \[\sum_{k=1}^{\infty} \frac{J_0(sk)}{k^2}\] and \[\sum_{k=1}^{\infty} \frac{J_1(sk)}{k}\] The theory demonstrates that the apparent oscillatory complexity of weighted Bessel series conceals a finite radical structure activated at multiples of $2\pi$ Each completed cycle contributes an explicit jump term, transforming infinite oscillatory sums into exact finite representations composed of polynomial components, radical terms, and transcendental functions. In particular, the derived identity for the $J_1$-weighted series is: \[\sum_{k=1}^{\infty} \frac{J_1(sk)}{k} = 1 - \frac{s}{4} + \frac{2}{s} \sum_{j=1}^{\left\lfloor \frac{s}{2\pi} \right\rfloor} \sqrt{s^2-(2\pi j)^2}\] This exact formula, valid for all s > 0, shows that the weighted $J_1$ series is governed by a finite sum of radical terms determined by completed $2\pi$-cycles, revealing a geometric structure underlying Bessel summations. The Binomial Law for Bessel Function Series therefore provides: Exact, factorized, and explicitly computable closed forms A deterministic alternative to purely numerical approximations A unified operator-based structure for weighted special-function series These results contribute to analytic summation theory and open perspectives for a broader structural classification of special-function series through finite jump representations.