Exact solutions to generalized Dunkl oscillator and its thermodynamic properties

In this work, based on the generalized Dunkl derivative in quantum mechanics we study the one-dimensional Schrödinger equation with a harmonic oscillator potential and obtain the energy eigenvalues. The principal thermodynamical properties including the Helmholtz free energy, mean energy and entropy are carried out. The effects of the Dunkl parameters on the thermodynamical quantities for even parity are discussed. The case of the odd parity can be easily obtained by substitution of the b rightarrow -band γ rightarrow -γ. All results in the limit case are reduced to ordinary statistical mechanics.