Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

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