Bargmann's versus of the quaternionic fractional Hankel transform

TL;DR

This paper explores the quaternionic fractional Hankel transform, extending the fractional Fourier transform to quaternions using hyperholomorphic Bargmann transforms, and establishes its fundamental properties like inversion and Plancherel identity.

Contribution

It introduces a quaternionic extension of the fractional Hankel transform via hyperholomorphic Bargmann transforms, providing foundational properties and analysis.

Findings

Derived inversion formula for the quaternionic fractional Hankel transform

Established Plancherel identity for the transform

Extended fractional Fourier analysis to quaternionic setting

Abstract

We investigate the quaternionic extension of the fractional Fourier transform on the real half-line leading to fractional Hankel transform. This will be handled \`a la Bargmann by means of hyperholomorphic second Bargmann transform for the slice Bergman space of second kind. Basic properties are derived including inversion formula and Plancherel identity.