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We show that multipartite generation functions can be written in terms of theBell polynomials (known as Fa\`a di Bruno's formula) and the Ruelle spectralfunctions, whose spectrum is encoded in the Patterson-Selberg function of thehyperbolic three-geometry. We derive an infinite-product formula for theChern-Simons partition functions and analyze appropriate q-series which leadsto the construction of knot invariants. With the help of the Ruelle spectralfunctions symmetric and modular properties in infinite-product structure can bedescribed.

Multipartite Generating Functions and Infinite Products for Quantum Invariants