The Fundamental Relation Between the Hurwitz Zeta Function and Bernoulli Polynomials: A Complete Spectral Proof

The Fundamental Relation Between the Hurwitz Zeta Function and Bernoulli Polynomials: A Complete Spectral Proof This work presents a novel and rigorous spectral proof of the classical identity connecting the Hurwitz zeta function and Bernoulli polynomials: ζ(1−k, x) = −Bₖ(x)/k for integers k ≥ 2. The proof is constructed within a specially designed Hilbert space framework, where the forward difference operator and its adjoint are explicitly computed and analyzed. The methodological approach combines techniques from functional analysis, spectral theory, and analytic number theory to establish this fundamental relationship through a fully detailed mathematical derivation. Key elements include the construction of the weighted Hilbert space H_x, the precise derivation of the adjoint operator Δ*, and the use of binomial expansion and summation-by-parts identities to bridge discrete and continuous aspects of the problem. The proof is supplemented by extensive numerical validation using high-precision computation (up to 50 decimal places), confirming the theoretical results with errors below 10⁻³⁰. This work not only provides a new perspective on a classical result but also introduces a spectral methodology that can be extended to other special function identities. It offers insights into the deep connections between operator theory, polynomial structures, and zeta functions, with potential applications in number theory, mathematical physics, and computational mathematics. The article is self-contained and includes all necessary preliminary definitions, detailed proofs, numerical verification data, and discussions on future research directions and generalizations.