This manuscript provides a complete, derivational treatment of the Meijer G-function, unifying its analytic, operator-theoretic, and spectral properties into a single structural framework. Starting from the Mellin-Barnes contour integral, the paper derives the associated generalized hypergeometric differential equation and proves the fundamental Mellin-transform identity. This identity rigorously characterizes the Meijer G-class as the exact inverse Mellin closure of finite gamma-function ratios. The framework then lifts this classical special function into operator theory. It constructs the induced multiplicative-convolution operator on the positive real half-line, conjugates it to additive convolution using logarithmic coordinates, and establishes exact compactness criteria for finite-window restrictions. This naturally leads to explicit trace expansions for Fredholm and regularized Fredholm determinants governed by cyclic integrals of the Meijer-G kernel. In parallel, the manuscript proves that heat traces and resolvent traces belonging to the Meijer-G class systematically generate spectral zeta functions via exact Mellin transforms. This provides explicit, gamma-ratio-driven formulas for zeta-regularized determinants under meromorphic continuation. Canonical examples, including the exponential, the algebraic resolvent kernel, and the modified Bessel K function, are derived to demonstrate the theory. Ultimately, the paper completely closes the Mellin, operator, Fredholm, and zeta-theoretic layers of the Meijer G-function into a single, cohesive theorem-proof architecture, requiring no external heuristic layers.
Andrew Kim (Sun,) studied this question.