  In this paper we briefly review the main idea of the localization technique
and its extension suitable in supersymmetric gauge field theory. We analyze the
partition function of the vector multiplets with supercharges and its blocks on
the even- and odd-dimensional spheres and squashed spheres. We exploit
so-called Fa\`a di Bruno's formula and show that multipartite partition
functions can be written in the form of expansion series of the Bell
polynomials. Applying the restricted specialization argument we show that
$q$-infinite-product representation of partition functions admits presentation
in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions.
