Algebraic properties of central Bell-based type 2 Bernoulli and Euler polynomials of complex variable

Waseem Ahmad Khan; Francesco Aldo Costabile; Khidir Shaib Mohamed; Ugur Duran; Abdulghani Muhyi; Azhar Iqbal; Wei Sin Koh; Waseem Ahmad Khan; Francesco Aldo Costabile; Khidir Shaib Mohamed; Ugur Duran; Abdulghani Muhyi; Azhar Iqbal; Wei Sin Koh

doi:10.3934/nhm.2026030

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 693-724. doi: 10.3934/nhm.2026030

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Algebraic properties of central Bell-based type 2 Bernoulli and Euler polynomials of complex variable

1.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
2.
Department of Mathematics and Computer Science, University of Calabria, Rende 87036, CS, Italy
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkiye
5.
Department of Mechatronics Engineering, Faculty of Engineering and Smart Computing, Modern Specialized University, Sana'a, Yemen
6.
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
7.
Faculty of Business and Communications, INTI International University, Nilai 71800, Negeri Sembilan, Malaysia

Received: 28 January 2026 Revised: 14 April 2026 Accepted: 15 April 2026 Published: 24 April 2026

Recently, by combining type 2 Bernoulli and Euler polynomials with the central Bell polynomials, the central Bell-based type 2 Bernoulli and Euler polynomials of order $ \alpha $ were considered, and many of their properties, formulas, and applications were investigated. The main aim of this work is to consider higher-order central Bell-based type 2 Bernoulli and Euler polynomials of complex variable, by which, both sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order $ \lambda $ are introduced by treating the imaginary and real components separately. Then, diverse summation formulas, differential formulas, addition formulas, and correlation formulas with new and existing old polynomials and numbers are derived in a systematic way. Also, several intriguing connections of sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order $ \lambda $ with the bivariate and one-variable central Bell polynomials, and the classical Stirling and central factorial numbers of the second kinds are investigated in detail. Moreover, the first few members of the new polynomials are provided by the lists, and the distributions of zeros of the new polynomials are illustrated by graphical representations, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.
- type 2 Euler polynomials,
- type 2 Bernoulli polynomials,
- central Bell polynomials,
- partial differential equations,
- differential equations,
- process innovation,
- sine and cosine polynomials
Citation: Waseem Ahmad Khan, Francesco Aldo Costabile, Khidir Shaib Mohamed, Ugur Duran, Abdulghani Muhyi, Azhar Iqbal, Wei Sin Koh. Algebraic properties of central Bell-based type 2 Bernoulli and Euler polynomials of complex variable[J]. Networks and Heterogeneous Media, 2026, 21(2): 693-724. doi: 10.3934/nhm.2026030

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Abstract

Recently, by combining type 2 Bernoulli and Euler polynomials with the central Bell polynomials, the central Bell-based type 2 Bernoulli and Euler polynomials of order $ \alpha $ were considered, and many of their properties, formulas, and applications were investigated. The main aim of this work is to consider higher-order central Bell-based type 2 Bernoulli and Euler polynomials of complex variable, by which, both sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order $ \lambda $ are introduced by treating the imaginary and real components separately. Then, diverse summation formulas, differential formulas, addition formulas, and correlation formulas with new and existing old polynomials and numbers are derived in a systematic way. Also, several intriguing connections of sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order $ \lambda $ with the bivariate and one-variable central Bell polynomials, and the classical Stirling and central factorial numbers of the second kinds are investigated in detail. Moreover, the first few members of the new polynomials are provided by the lists, and the distributions of zeros of the new polynomials are illustrated by graphical representations, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.

References

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