Some properties of Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials and their fractional extension

Mesfer H. Alqahtani; Abdulghani Muhyi; Amel Touati; Muntasir Suhail; L. M. Abdalgadir; Khaled Aldwoah; Amer Alsulami; Mesfer H. Alqahtani; Abdulghani Muhyi; Amel Touati; Muntasir Suhail; L. M. Abdalgadir; Khaled Aldwoah; Amer Alsulami

doi:10.3934/math.2026512

AIMS Mathematics

2026, Volume 11, Issue 5: 12449-12477. doi: 10.3934/math.2026512

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Research article

Some properties of Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials and their fractional extension

1.
Department of Mathematics, University College of Umluj, University of Tabuk, Tabuk 48322, Saudi Arabia
2.
Department of Mathematics, Hajjah University, Hajjah, Yemen
3.
Department of Mathematics, Faculty of Science, Northern Border University, Arar 91431, Saudi Arabia
4.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
5.
Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
6.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
7.
Department of Mathematics, Turabah University College, Taif University, Taif, Saudi Arabia

Received: 25 February 2026 Revised: 21 April 2026 Accepted: 23 April 2026 Published: 06 May 2026
MSC : 11B68, 11B73, 26A33, 33C47, 33E20

This study presents a new, generalized family of special polynomials. These polynomials are designated as the Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials. Based on their generating function, the corresponding series expansions and summation formulas are obtained. Furthermore, determinant representation and several differential and integral representations of these polynomials are derived. The fractional extension of this polynomial family is explored, resulting in the formulation of several associated identities. Finally, the study utilizes Mathematica to deliver zero distributions and graphical representations.
Citation: Mesfer H. Alqahtani, Abdulghani Muhyi, Amel Touati, Muntasir Suhail, L. M. Abdalgadir, Khaled Aldwoah, Amer Alsulami. Some properties of Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials and their fractional extension[J]. AIMS Mathematics, 2026, 11(5): 12449-12477. doi: 10.3934/math.2026512

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Abstract

This study presents a new, generalized family of special polynomials. These polynomials are designated as the Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials. Based on their generating function, the corresponding series expansions and summation formulas are obtained. Furthermore, determinant representation and several differential and integral representations of these polynomials are derived. The fractional extension of this polynomial family is explored, resulting in the formulation of several associated identities. Finally, the study utilizes Mathematica to deliver zero distributions and graphical representations.

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