A family of trivariate Gould-Hopper-Bell-Apostol-Type polynomials: Construction and properties

Rabeb Sidaoui; Abdulghani Muhyi; Khaled Aldwoah; Ayman Alahmade; Mohammed Rabih; Amer Alsulami; Khidir Mohamed; Rabeb Sidaoui; Abdulghani Muhyi; Khaled Aldwoah; Ayman Alahmade; Mohammed Rabih; Amer Alsulami; Khidir Mohamed

doi:10.3934/math.2025594

AIMS Mathematics

2025, Volume 10, Issue 6: 13251-13277. doi: 10.3934/math.2025594

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A family of trivariate Gould-Hopper-Bell-Apostol-Type polynomials: Construction and properties

1.
Department of Mathematics, College of Science, University of Ha'il, 55473 Ha'il, Saudi Arabia
2.
Department of Mathematics, Hajjah University, Hajjah, Yemen
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
4.
Department of Mathematics, College of Science, Taibah University, Medinah 42353, Saudi Arabia
5.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
6.
Department of Mathematics, Turabah University College, Taif University, Taif, Saudi Arabia

Received: 21 April 2025 Revised: 23 May 2025 Accepted: 29 May 2025 Published: 09 June 2025
MSC : 11B68, 11B73, 11B83, 05A15, 33C47

In this study, we define and investigate a new hybrid family of special polynomials, named the trivariate Gould-Hopper-Bell-Apostol-type polynomials. By applying the monomiality principle, we construct their generating function, derive series representations, identify quasi-monomial operators, and formulate the corresponding differential equation. We also establish summation formulas and integral/differential representations. Furthermore, we investigate specific members of this family, including Gould-Hopper-Bell-Apostol-Bernoulli, Gould-Hopper-Bell-Apostol-Euler, and Gould-Hopper-Bell-Apostol-Genocchi polynomials, revealing analogous results for each. Finally, we utilize Mathematica for computational exploration, examining zero distributions and graphical representations.
- Apostol-type polynomials,
- Gould-Hopper polynomials,
- Bell polynomials,
- monomiality principle,
- generating function,
- differential equations,
- zero distributions
Citation: Rabeb Sidaoui, Abdulghani Muhyi, Khaled Aldwoah, Ayman Alahmade, Mohammed Rabih, Amer Alsulami, Khidir Mohamed. A family of trivariate Gould-Hopper-Bell-Apostol-Type polynomials: Construction and properties[J]. AIMS Mathematics, 2025, 10(6): 13251-13277. doi: 10.3934/math.2025594

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Abstract

In this study, we define and investigate a new hybrid family of special polynomials, named the trivariate Gould-Hopper-Bell-Apostol-type polynomials. By applying the monomiality principle, we construct their generating function, derive series representations, identify quasi-monomial operators, and formulate the corresponding differential equation. We also establish summation formulas and integral/differential representations. Furthermore, we investigate specific members of this family, including Gould-Hopper-Bell-Apostol-Bernoulli, Gould-Hopper-Bell-Apostol-Euler, and Gould-Hopper-Bell-Apostol-Genocchi polynomials, revealing analogous results for each. Finally, we utilize Mathematica for computational exploration, examining zero distributions and graphical representations.

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