On the Bessel function and $n$-dimensional Hankel transform with Bicomplex arguments and coherent states

TL;DR

This paper introduces and analyzes bicomplex Bessel functions, explores their properties, and applies an $n$-dimensional bicomplex Hankel transform to solve PDEs and extend coherent states.

Contribution

It presents the first comprehensive study of bicomplex Bessel functions, defines an $n$-dimensional bicomplex Hankel transform, and extends coherent states using these functions.

Findings

Bicomplex Bessel functions satisfy recurrence, integral, and differential relations.

The $n$-dimensional bicomplex Hankel transform is an isomorphism between function spaces.

New bicomplex coherent states are constructed fulfilling key quantum conditions.

Abstract

In this work, we introduce bicomplex Bessel function and analyze its region of convergence. Important properties of the bicomplex Bessel function, such as recurrence relations, integral representations, differential relations are explored. Moreover a differential equation satisfied by the bicomplex Bessel function is established. Furthermore, we investigate bicomplex holomorphicity and discuss its asymptotic behavior. Finally, we define $n$-dimensional bicomplex Hankel transformation by using bicomplex Bessel function and show that it is an isomorphism between two suitably defined function spaces. The application of the $n$-dimensional bicomplex Hankel transform has been effectively demonstrated by solving some partial differential equations. Additionally, a new extension of coherent states is built based on the use of the bicomplex Bessel function and demonstrate that these states fulfill the conditions of normalizability, continuity and the resolution of unity.