Advancements in $ q $-Hermite-Appell polynomials: a three-dimensional exploration

Mohra Zayed; Shahid Ahmad Wani; William Ramírez; Clemente Cesarano; Mohra Zayed; Shahid Ahmad Wani; William Ramírez; Clemente Cesarano

doi:10.3934/math.20241303

AIMS Mathematics

2024, Volume 9, Issue 10: 26799-26824. doi: 10.3934/math.20241303

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Advancements in $ q $-Hermite-Appell polynomials: a three-dimensional exploration

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India
3.
Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58, 55-66, Barranquilla 080002, Colombia
4.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 39, Rome 00186, Italy

Received: 29 April 2024 Revised: 15 July 2024 Accepted: 26 July 2024 Published: 14 September 2024
MSC : 33E20, 33C45, 33B10, 33E30, 11T23

In this research, we leverage various $ q $-calculus identities to introduce the notion of $ q $-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $ q $-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, $ q $-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of $ q $-calculus.
- special polynomials,
- $ q $-calculus,
- monomiality principle,
- explicit form,
- operational connection,
- determinant form,
- symmetric identities,
- summation formulae
Citation: Mohra Zayed, Shahid Ahmad Wani, William Ramírez, Clemente Cesarano. Advancements in $ q $-Hermite-Appell polynomials: a three-dimensional exploration[J]. AIMS Mathematics, 2024, 9(10): 26799-26824. doi: 10.3934/math.20241303

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Abstract

In this research, we leverage various $ q $-calculus identities to introduce the notion of $ q $-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $ q $-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, $ q $-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of $ q $-calculus.

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