Inverse Power Fourier-Bessel Series: Binomial Formulas, Integral Extensions, and Algebraic Example

Abstract / Description: This repository presents a unified framework for inverse power Fourier-Bessel series of the type \₊=₁^ J_ (s k) kᵐ. \ The originality of this work relies on three complementary aspects: Exact binomial structure for 0 2 Using periodized Bernoulli polynomials, we construct integral representations that provide a rigorous analytic continuation of the binomial formulas beyond the fundamental period. This approach captures periodic jumps and discontinuities analytically, ensuring continuity across multiples of 2 Algebraic simplifications for key arguments For special values such as s = n, 2, 3, 4 the series reduce to simplified algebraic forms involving radicals, critical angles, and zeta constants, highlighting the underlying arithmetic richness. Scientific impact and originality: The integral representations serve as exact generating functions for the binomial structures. Residual terms possess strict binomial form controlled by central coefficients, isolating rational contributions and allowing compact exact expressions. The integral extension provides a global analytic continuation, unifying the odd (J₁) and even (J₀) Bessel families via differential relations. The approach bridges harmonic analysis, combinatorial structures, and analytic continuation, producing exact analytic objects beyond classical numerical approximations.