Analytic Umbral Transmutations and Bessel Moments

We apply the recently proposed analytic extension of formal indicial umbral calculus to the evaluation and structural interpretation of Bessel moments, replacing formal symbolic constructions with Mellin–Barnes analytic transmutations. The classical umbral representation of J0 converts products of Bessel functions into Gaussian integrals involving sums of independent symbolic operators. This formal mechanism may reproduce correct identities in suitable convergence chambers, but it may also lead to non-admissible hypergeometric expansions at the physically relevant parameter values. The cubic moment already exhibits this difficulty: the formal Appell F4 expansion associated with the equilateral case lies outside its domain of convergence. We address this obstruction by replacing the formal expansion with Mellin–Barnes representations of the corresponding umbral pairings. In this formulation, Ramanujan’s Master Theorem selects the analytic ground state associated with a Bessel product. The factorisation J03=J0J02 fuses the elementary Bessel state with the square state and gives the cubic moment as a one-dimensional Meijer–Barnes function. The same mechanism yields a scaled cubic formula and a fourth-moment Meijer–Barnes representation whose residues give a convergent harmonic-number expansion. The fifth moment marks the first higher-rank case: the natural grouping J05=J02J02J0 leads to a bivariate Barnes transmutation rather than to an ordinary Meijer G-function. Finally, real powers J0α, α>2, are interpreted through Mellin-selected ground states, which need not reduce to finite Gamma products. Thus, Bessel moments provide a concrete hierarchy of analytic umbral representations, from rank-one Meijer–Barnes functions to higher-rank Barnes structures, and distinguish the global analytic meaning of an umbral construction from the local convergence of its residue expansions.