Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials

Abstract

<abstract><p>In this study, we develop various features in special polynomials using the principle of monomiality, operational formalism, and other qualities. By utilizing the monomiality principle, new outcomes can be achieved while staying consistent with past knowledge. Furthermore, an explicit form satisfied by these polynomials is also derived. The emphasis of this study is to introduce the degenerate multidimensional Hermite polynomials (DMVHP) denoted as $ \mathbb{H}^{[r]}_n(j_1, j_2, j_3, \cdots, j_r; \vartheta) $, which are closely related to the classical Hermite polynomials and are a significant class of orthogonal polynomials. The fundamental properties, such as symmetric identities for these polynomials are also established. An operational framework is also established for these polynomials.</p></abstract>