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We construct nonlinear coherent states or f-deformed coherent states for anonpolynomial nonlinear oscillator which can be considered as placed in themiddle between the harmonic oscillator and the isotonic oscillator (Cari\~nenaJ F et al, J. Phys. A: Math. Theor. 41, 085301 (2008)). The deformedannihilation and creation operators which are required to construct thenonlinear coherent states in the number basis are obtained from the solution ofthe Schr\"odinger equation. Using these operators, we construct generalizedintelligent states, nonlinear coherent states, Gazeau-Klauder coherent statesand the even and odd nonlinear coherent states for this newly solvable system.We also report certain nonclassical properties exhibited by these nonlinearcoherent states. In addition to the above, we consider position dependent massSchr\"odinger equation associated with this solvable nonlinear oscillator andconstruct nonlinear coherent states, Gazeau-Klauder coherent states and theeven and odd nonlinear coherent states for it. We also give explicitexpressions of all these nonlinear coherent states by considering a massprofile which is often used for studying transport properties insemiconductors.

On the generalized intelligent states and certain related nonclassical states of a quantum exactly solvable nonlinear oscillator