  Revisiting the Coon amplitude, a deformation of the Veneziano amplitude with
a logarithmic generalization of linear Regge trajectories, we scrutinize its
potential origins in a worldsheet theory by proposing a definition of its
$q$-deformation through the integral representation of the $q$-beta function.
By utilizing $q$-deformed commutation relations and vertex operators, we derive
the Coon amplitude within the framework of the dual resonance model. We extend
this to the open-string context by $q$-deforming the Lie algebra
$\mathfrak{su}(1,1)$, resulting in a well-defined $q$-deformed open superstring
amplitude. We further demonstrate that the $q$-prefactor in the Coon amplitude
arises naturally from the property of the $q$-integral. Furthermore, we find
that two different types of $q$-prefactors, corresponding to different
representations of the same scattering amplitude, are essentially the same by
leveraging the properties of $q$-numbers. Our findings indicate that the
$q$-deformed string amplitude defines a continuous family of amplitudes,
illustrating how string amplitudes with a finite $\alpha^\prime$ uniquely flow
to the amplitudes of scalar scattering in field theory at energy scale
$\Lambda$ as $q$ changes from $1$ to $0$. This happens without the requirement
of an $\alpha^\prime$ expansion, presenting a fresh perspective on the
connection between string and field theories.
