---
abstract: 'We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials consatitute orthogonal families of vectors in ambient quaternionic $L^2$-spaces. Using these polynomials, we then define regular and anti-regular subspaces of these $L^2$-spaces, the associated reproducing kernels and the ensuing quaternionic coherent states.'
address:
- '$^{1}$ Department of Computer Science and Software Engineering, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada. '
- '$^2$ Department of mathematics and Statistics, Concordia University, Montreal, Quebec, H3G 1M8, Canada'
author:
- 'K. Thirulogasanthar$^{1}$ and S. Twareque Ali$^{2}$'
title: Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated Coherent states
---

[^1]

Introduction {#sec_intro}
============

Building quantum mechanics on quaternionic Hilbert spaces has been a much studied problem for many years (see, for example, [@Ad] and the many references cited therein). Associated to this problem is that of building appropriate families of coherent states on quaternionic Hilbert spaces. The fact that the analogues of the usual canonical coherent states cannot be built using a group theoretical argument in the case of quaternions, has been elaborated in [@AdMil]. On the other hand, analogues of such 