Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials

Mohra Zayed; Shahid Wani; Mohra Zayed; Shahid Wani

doi:10.3934/math.20231575

AIMS Mathematics

2023, Volume 8, Issue 12: 30813-30826. doi: 10.3934/math.20231575

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Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials

Mohra Zayed ¹,
Shahid Wani ^{2
,
,}

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed) University (SIU), Pune, Maharashtra, India

Received: 03 May 2023 Revised: 01 August 2023 Accepted: 16 August 2023 Published: 15 November 2023
MSC : 11T23, 33B10, 33C45, 33E20, 33E30

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.
- degenerate special polynomials,
- Hermite polynomials,
- monomiality principle,
- explicit form,
- operational formalism,
- symmetric identities
Citation: Mohra Zayed, Shahid Wani. Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials[J]. AIMS Mathematics, 2023, 8(12): 30813-30826. doi: 10.3934/math.20231575

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Abstract

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.

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