• Corpus ID: 215828207

Analytic aspects of exceptional Hermite polynomials and associated minimal surfaces

@article{Chalifour2020AnalyticAO,
  title={Analytic aspects of exceptional Hermite polynomials and associated minimal surfaces},
  author={V. Chalifour and Alfred Michel Grundland},
  journal={arXiv: Mathematical Physics},
  year={2020}
}

• V. Chalifour, A. Grundland
• Published 20 April 2020
• Mathematics
• arXiv: Mathematical Physics

The main aim of this paper is to construct minimal surfaces associated with exceptional Hermite polynomials. That is, we derive an ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the analytical general solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is completed by non-polynomial solutions, and that the exceptional Hermite polynomials of odd and even degrees correspond respectively to the odd and even powers of a series that is part of the general solution. The paper contains a detailed exposition of the theory of exceptional Hermite polynomials as well as an exposition of the main tool used to construct their associated minimal surfaces obtained through the Enneper-Weierstrass immersion representation. We present 3D numerical displays of these surfaces. Several examples of minimal surfaces are included as illustrations of the presented theoretical results.