Several characterizations of bivariate quantum-Hermite-Appell Polynomials and the structure of their zeros

Mohra Zayed; Taghreed Alqurashi; Shahid Ahmad Wani; Cheon Seoung Ryoo; William Ramírez; Mohra Zayed; Taghreed Alqurashi; Shahid Ahmad Wani; Cheon Seoung Ryoo; William Ramírez

doi:10.3934/math.2025507

AIMS Mathematics

2025, Volume 10, Issue 5: 11184-11207. doi: 10.3934/math.2025507

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Several characterizations of bivariate quantum-Hermite-Appell Polynomials and the structure of their zeros

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Mathematics Department, Faculty of Science, Al-Baha University, 65779-7738 Albaha City, Kingdom of Saudi Arabia
3.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India
4.
Department of Mathematics, Hannam University, Daejeon 34430, South Korea
5.
Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla 080002, Colombia
6.
Section of Mathematics International Telematic University Uninettuno, Rome 00186, Italy

Received: 06 March 2025 Revised: 16 April 2025 Accepted: 22 April 2025 Published: 15 May 2025
MSC : 33E20, 33C45, 33B10, 33E30, 11T23

This paper investigated the fundamental characteristics and uses of a new class of bivariate quantum-Hermite-Appell polynomials. The series representation and generating relation for these polynomials were derived. Also, a determinant representation for these polynomials was derived. Further, important mathematical characteristics were derived, such as $ q $-recurrence relations and $ q $-difference equations. These polynomials' numerical features were methodically examined, providing information on their computational possibilities and the framework of their zeros. A coherent framework was established by extending the study to related families, such as quantum-Hermite Bernoulli, quantum-Hermite Euler, and quantum-Hermite Genocchi polynomials. These discoveries enhance the knowledge of quantum polynomials and their relationships to classical and contemporary special functions.
- special functions,
- quantum calculus,
- explicit form,
- operational connection,
- structure of zeros,
- determinant form
Citation: Mohra Zayed, Taghreed Alqurashi, Shahid Ahmad Wani, Cheon Seoung Ryoo, William Ramírez. Several characterizations of bivariate quantum-Hermite-Appell Polynomials and the structure of their zeros[J]. AIMS Mathematics, 2025, 10(5): 11184-11207. doi: 10.3934/math.2025507

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Abstract

This paper investigated the fundamental characteristics and uses of a new class of bivariate quantum-Hermite-Appell polynomials. The series representation and generating relation for these polynomials were derived. Also, a determinant representation for these polynomials was derived. Further, important mathematical characteristics were derived, such as $ q $-recurrence relations and $ q $-difference equations. These polynomials' numerical features were methodically examined, providing information on their computational possibilities and the framework of their zeros. A coherent framework was established by extending the study to related families, such as quantum-Hermite Bernoulli, quantum-Hermite Euler, and quantum-Hermite Genocchi polynomials. These discoveries enhance the knowledge of quantum polynomials and their relationships to classical and contemporary special functions.

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