Constructing Barut-Girardello coherent states for the isotonic oscillator in the DOOT approach

In this work, we study the quantum system of the isotonic oscillator from the perspective of the diagonal operator ordering technique (DOOT). Within this framework, we construct the associated Barut-Girardello and Gazeau-Klauder coherent states. We examine their mathematical properties using reproducing kernels and compute the expectation values of observables that characterize the system and its relevant physical features. Further, we perform the quantization of main classical variables in the complex plane. Then, by exploring the thermal behavior of the physical system in the constructed coherent states, we analyze the properties of mixed states described by a canonical density operator. We also obtain the corresponding Glauber-Sudarshan P-representation.