Para-Bose oscillator algebras of odd orders: x-representations and Wigner functions for coherent and cat states and their photon-added and photon-subtracted counterparts

The expansion coefficients for the Glauber-type coherent states | z ⟩ λ of a para-Bose algebra of odd order P = 2 λ + 1 are expressed in terms of the Laguerre polynomials of order λ in | z | 2 . It allows us to use one of the generating functions of the Laguerre polynomials to obtain the x -representation of the coherent states in terms of the J -Bessel functions. This, in turn, makes it possible to evaluate x -representations of the even and odd cat states as well as the m -photon-added and m -photon-subtracted coherent and cat states of the para-Bose oscillator algebra of odd order P = 2 λ + 1 . Finally, with the help of these x -representations, we are able to examine Wigner functions and observe the non-classical behaviors via negative values for the probability distribution functions in phase space.