Exploring differential equations and fundamental properties of Generalized Hermite-Frobenius-Genocchi polynomials

Awatif Muflih Alqahtani; Shahid Ahmad Wani; William Ramírez; Awatif Muflih Alqahtani; Shahid Ahmad Wani; William Ramírez

doi:10.3934/math.2025125

AIMS Mathematics

2025, Volume 10, Issue 2: 2668-2683. doi: 10.3934/math.2025125

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Exploring differential equations and fundamental properties of Generalized Hermite-Frobenius-Genocchi polynomials

1.
Department of Mathematics, Shaqra University, Riyadh, Saudi Arabia; aalqhtani@su.edu.sa
2.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India; shahid.wani@sitpune.edu.in
3.
Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla 080002, Colombia; wramirez4@cuc.edu.co
4.
Section of Mathematics International Telematic University Uninettuno, Rome 00186, Italy

Received: 06 September 2024 Revised: 07 January 2025 Accepted: 10 January 2025 Published: 14 February 2025
MSC : 33E20, 33C45, 33B10, 33E30, 39A14, 45J05, 11T23

This study introduces an innovative framework for generalized Hermite-Frobenius-Genocchi polynomials in two variables, parameterized by a single variable. The focus is on providing a comprehensive characterization of these polynomials through various mathematical tools, including generating functions, series expansions, and summation identities that uncover their essential properties. The work extends to the derivation of recurrence relations, the investigation of shift operators, and the formulation of multiple types of differential equations. In particular, the study delves into integro-differential and partial differential equations, employing a factorization technique to develop different forms and solutions. This multifaceted approach not only enhances our understanding of these polynomials, but also lays the groundwork for their further exploration in diverse areas of mathematical research.
Citation: Awatif Muflih Alqahtani, Shahid Ahmad Wani, William Ramírez. Exploring differential equations and fundamental properties of Generalized Hermite-Frobenius-Genocchi polynomials[J]. AIMS Mathematics, 2025, 10(2): 2668-2683. doi: 10.3934/math.2025125

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Abstract

This study introduces an innovative framework for generalized Hermite-Frobenius-Genocchi polynomials in two variables, parameterized by a single variable. The focus is on providing a comprehensive characterization of these polynomials through various mathematical tools, including generating functions, series expansions, and summation identities that uncover their essential properties. The work extends to the derivation of recurrence relations, the investigation of shift operators, and the formulation of multiple types of differential equations. In particular, the study delves into integro-differential and partial differential equations, employing a factorization technique to develop different forms and solutions. This multifaceted approach not only enhances our understanding of these polynomials, but also lays the groundwork for their further exploration in diverse areas of mathematical research.

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