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We analyze the characteristic series, the $KO$ series and the seriesassociated with the Witten genus, and their analytic forms as the $q$-analogsof classical special functions (in particular $q$-analog of the beta integraland the gamma function). $q$-series admit an analytic interpretation in termsof the spectral Ruelle functions, and their relations to appropriate ellipticmodular forms can be described. We show that there is a deep correspondencebetween the characteristic series of the Witten genus and $KO$ characteristicseries, on one side, and the denominator identities and characters of $N=2$superconformal algebras, and the affine Lie (super)algebras on the other. Werepresent the characteristic series in the form of double series using theHecke-Rogers modular identity.

Elliptic Genera and Characteristic $q$-Series of Superconformal Field Theory