The coupling between the quantum generating functions of a field theory and corresponding formal power series, associated with dimensions of chains and homologies of suitable Lie algebras, is explored. This interrelation is further analyzed in view to its application to the Casimir effect for partition functions of minimal three-dimensional gravities, in space-times which are asymptotic to, respectively, AdS 3 , pure N = 1 supergravity, two-variable elliptic genera of an hypersurface, and

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... te intersections. All these examples are here viewed as straightforward applications of the machinery of modular forms and spectral functions to the one-loop contributions of quantum partition functions (e.g. the Casimir effect) which are represented by means of formal power series.