Using the technique of integration within an ordered product of nonlinear Bose operators and by composing the nonlinear coherent state's over-completeness relation, we construct the corresponding P -representation theory. The generalized P -representation for some nonlinear Bose operators can be established.
Based on the newly constructed overcompleteness relation of the nonlinear coherent states (NLCSs) and corresponding 𝒫-representation we derive: (1) a new antinormally ordered expansion of density matrices; and (2) the corresponding positive 𝒫-representation. Our derivation employs the technique of integration within an ordered product of nonlinear Bose operator functions. A relationship between the positive 𝒫-representation and the generalized characteristic function in the context of NLCSs is also obtained.