  The Tsallis $q$-exponential function $e_q(x) = (1+(1-q)x)^{\frac{1}{1-q}}$ is
found to be associated with the deformed oscillator defined by the relations
$\left[N,a^\dagger\right] = a^\dagger$, $[N,a] = -a$, and
$\left[a,a^\dagger\right] = \phi_T(N+1)-\phi_T(N)$, with $\phi_T(N) =
N/(1+(q-1)(N-1))$. In a Bargmann-like representation of this deformed
oscillator the annihilation operator $a$ corresponds to a deformed derivative
with the Tsallis $q$-exponential functions as its eigenfunctions, and the
Tsallis $q$-exponential functions become the coherent states of the deformed
oscillator. When $q = 2$ these deformed oscillator coherent states correspond
to states known variously as phase coherent states, harmonious states, or
pseudothermal states. Further, when $q = 1$ this deformed oscillator is a
canonical boson oscillator, when $1 < q < 2$ its ground state energy is same as
for a boson and the excited energy levels lie in a band of finite width, and
when $q \longrightarrow 2$ it becomes a two-level system with a nondegenerate
ground state and an infinitely degenerate excited state.
