Positive Definiteness of Products of Normalized Bessel Functions

We study positive definiteness on the real line for products of normalized Bessel functions. We obtain an exact classification through a real-integral sign criterion in the nonclassical parameter region, prove a monotone parameter transport and an analytic boundary, and analyze the signed spectral density using a third-order Meijer differential equation and disconjugacy. We derive asymptotic expansions near the lemniscatic corner and report rigorously certified global extremal properties within the scope described in the manuscript.