Connection between coherent states and some integrals and integral representations

TL;DR

This paper explores the connection between coherent states in quantum mechanics and integrals involving special functions, introducing a novel method to derive and evaluate integral representations using the diagonal operators ordering technique.

Contribution

It presents a new approach linking coherent states to integral representations, enabling the calculation of previously unknown integrals involving special functions.

Findings

Derived new integral representations involving Meijer's G- and hypergeometric functions.

Provided a method to evaluate integrals using properties of the diagonal operators ordering technique.

Expanded the set of solvable integrals involving special functions.

Abstract

The paper presents an interesting mathematical feedback between the formalism of coherent states and the field of integrals and integral representations involving special functions. This materializes through an easy and fast method to calculate integrals or integral representations of different functions, expressible by means of Meijer's G-, as well as hypergeometric generalized functions. The feedback starts from a fundamental integral that comes from the decomposition of the unity operator in the language of coherent states from quantum mechanics. In this way, integrals and integral representations are obtained, some that do not appear in the literature, and others already known, which can be verified by orthodox methods. All calculations are made using the properties of the diagonal operators ordering technique (DOOT), a relatively new technique of normal ordering of the creation and annihilation operators in quantum mechanics. The paper contributes to increasing the number of solvable integrals involving special functions.