  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 a $q$-deformation of the Veneziano amplitude for all $q \geq 0$
and discover a new transcendental structure in its low-energy expansion. We
show that there is no analogous $q$-deformation of the Virasoro amplitude.
