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