On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential

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\[N,a^daggerright\] = a^dagger, [N,a] = -a, and left\[a,a^daggerright\] = φ_T(N+1)-φ_T(N), with φ_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.