We analyze so-called generalized Veneziano and generalized Virasoro amplitudes. Under some physical assumptions, we find that their spectra must satisfy an over-determined set of non-linear recursion relations. The recursion relation for the generalized Veneziano amplitudes can be solved analytically and yields a two-parameter family which includes the Veneziano amplitude, the one-parameter family of Coon amplitudes, and a larger two-parameter family of amplitudes with an infinite tower of spins at each mass level. In the generalized Virasoro case, the only consistent solution is the string spectrum.
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A<sc>bstract</sc> -
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A<sc>bstract</sc> We detail the properties of the Veneziano, Virasoro, and Coon amplitudes. These tree-level four-point scattering amplitudes may be written as infinite products with an infinite sequence of simple poles. Our approach for the Coon amplitude uses the mathematical theory of
q -analysis. We interpret the Coon amplitude as aq -deformation of the Veneziano amplitude for allq ≥ 0 and discover a new transcendental structure in its low-energy expansion. We show that there is no analogousq -deformation of the Virasoro amplitude.
