Analytic Bernoulli Functions: Correspondence with Hermite Polynomials

Abstract: We establish an operator--theoretic correspondence between periodic Bernoulli kernels and Hermite polynomials, framed through the umbral calculus and a quantum analogy. Starting from the analytic master function $F^\ast$, the periodic Hilbert transform appears as a $\pi/2$ symplectic rotation, while Jacobi matrices reproduce the oscillator ladder. Umbral operational rules extend this structure across complex order, and Weyl algebra lifting together with the Weil representation explains the shared spectrum of Bernoulli and Hermite families. Analytically, this chain connects Clausen functions to Bernoulli kernels, to polylogarithms, to the Hurwitz zeta function, and ultimately to the Lerch transcendent, embedding the umbral framework into the classical landscape of special functions. This perspective clarifies why odd zeta values arise in Bernoulli integrals and unifies trigonometric and Gaussian worlds within a coherent operator framework.