Bargmann transform and statistical properties for nonlinear coherent states of the isotonic oscillator

Abstract We construct a new class of nonlinear coherent states for the isotonic oscillator by replacing the factorial of the coefficients z n / n ! ${z}^{n}/\sqrt{n!}$ of the canonical coherent states by the factorial x n γ ! = x 1 γ . x 2 γ … x n γ ${x}_{n}^{\gamma }!={x}_{1}^{\gamma }.{x}_{2}^{\gamma }\dots {x}_{n}^{\gamma }$ with x 0 γ = 0 ${x}_{0}^{\gamma }=0$ , where x n γ ${x}_{n}^{\gamma }$ is a sequence of positive numbers and γ is a positive real parameter. This also leads to the construction of a Bargmann-type integral transform which will allow us to find some integral transforms for orthogonal polynomials. The statistics of our coherent states will also be considered by the calculus of one called Mandel parameter. The squeezing phenomenon was also discussed.