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In our preceding work, a class of -quantum nonlinear coherent states, i.e., the eigenstates of the powers (of the annihilation operator of -oscillators, are introduced. In this paper, we introduce a new kind of higher-order squeezing and an antibunching effect. The quantum statistical properties of the states are studied. The result shows that the -th order ; squeezing effects exist in all of the states when is even. There is the antibunching effect in all of the states.

A new kind of k-quantum nonlinear coherent states, i.e., the k eigenstates of the k-th power of the generalized annihilation operator of f-oscillators, are obtained and their properties are discussed. The completeness of the k states is investigated. An alternative method to construct them is proposed. It is shown that these states may form a complete Hilbert space, and all of them can be generated by a linear superposition of k Roy-type nonlinear coherent states. Physically, they can be generated by a linear superposition of the time-dependent Roy-type nonlinear coherent states at different instants.