# Nonlinear coherent states associated with a measure on the positive real half line

## TL;DR

This paper introduces a new class of nonlinear coherent states derived from 2D complex orthogonal polynomials linked to a specific measure on the positive real line, expanding the mathematical framework of quantum state representations.

## Contribution

It develops a novel construction of nonlinear coherent states based on a measure on the positive real line and explores their transform and polynomial basis representations.

## Key findings

- Constructed generalized nonlinear coherent states from 2D orthogonal polynomials.
- Established a coherent states transform related to these states.
- Provided a polynomial basis realization of quantum states.

## Abstract

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated coherent states transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this construction with the example of the measure $r^\beta e^{-r}dr$, which leads to a new generalization of the true-polyanalytic Bargmann transform.