Investigating Multidimensional Degenerate Hybrid Special Polynomials and Their Connection to Appell Sequences: Properties and Applications

TL;DR

This paper introduces degenerate multidimensional Hermite-based Appell polynomials, exploring their properties, connections to classical polynomials, and potential applications through an operational framework.

Contribution

It presents a novel class of degenerate multidimensional Hermite-based Appell polynomials and investigates their properties and connections to classical polynomials.

Findings

Introduction of degenerate multidimensional Hermite-based Appell polynomials

Derivation of symmetric identities for these polynomials

Establishment of an operational framework for further study

Abstract

This paper explores the operational principles and monomiality principle that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as ${}_{\mathbb{H}}\mathbb{A}^{[r]}_n(l_1, l_2, l_3, \dots, l_r; \vartheta)$. These DMHAP form essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further.