  We analyze the characteristic series, the $KO$ series and the series
associated with the Witten genus, and their analytic forms as the $q$-analogs
of classical special functions (in particular $q$-analog of the beta integral
and the gamma function). $q$-series admit an analytic interpretation in terms
of the spectral Ruelle functions, and their relations to appropriate elliptic
modular forms can be described. We show that there is a deep correspondence
between the characteristic series of the Witten genus and $KO$ characteristic
series, on one side, and the denominator identities and characters of $N=2$
superconformal algebras, and the affine Lie (super)algebras on the other. We
represent the characteristic series in the form of double series using the
Hecke-Rogers modular identity.
