(p,q)-integration is defined for the (p,q)-oscillator. This is used to prove a completeness relation for the coherent states of the (p,q)-oscillator. The (p,q)-analogue of the Bargmann-Fock representation is also discussed.
Using the completeness relation composed of the coherent state and of the eigenket of bosonic creation operator, we establish a one-to-one correspondence between the z -transform and the quantum-mechanical transform from the representation by number states to the representation by coherent states (Bargmann representation). In this way, the quantum-mechanical version of the various properties of z -transform are obtained and the operators for embodying these properties in the Fock space are derived, which may find applications in quantum states engineering.
A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmann's scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.