A new class of generalized Apostol–type Frobenius–Euler polynomials

Letelier Castilla; William Ramírez; Clemente Cesarano; Shahid Ahmad Wani; Maria-Fernanda Heredia-Moyano; Letelier Castilla; William Ramírez; Clemente Cesarano; Shahid Ahmad Wani; Maria-Fernanda Heredia-Moyano

doi:10.3934/math.2025167

AIMS Mathematics

2025, Volume 10, Issue 2: 3623-3641. doi: 10.3934/math.2025167

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A new class of generalized Apostol–type Frobenius–Euler polynomials

1.
I.E.E. Normal Superior Nuestra Señora de Fátima. Sabanagrande, Colombia
2.
Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58, 55-66, 080002 Barranquilla, Colombia
3.
Section of Mathematics International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy
4.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed) University, Pune, India

Received: 20 December 2024 Revised: 12 February 2025 Accepted: 14 February 2025 Published: 25 February 2025
MSC : 11B68, 11B83, 11B39, 05A19

The paper presents a new type of generalized Apostol-type Frobenius–Euler polynomials and numbers with specific order $ \kappa $ and level $ m $. We establish fundamental identities and properties using generating function techniques, such as summation formulas, differential and integral relations, and addition theorems. Additionally, we explore the connections between these polynomials and the Stirling numbers of the second kind, as well as other polynomial families. Lastly, we derive a differential equation and a recurrence relation for these new classes of polynomials. Finally, we show applications that can be obtained using these polynomials where the graphs of the zero functions and the meshes are displayed.
Citation: Letelier Castilla, William Ramírez, Clemente Cesarano, Shahid Ahmad Wani, Maria-Fernanda Heredia-Moyano. A new class of generalized Apostol–type Frobenius–Euler polynomials[J]. AIMS Mathematics, 2025, 10(2): 3623-3641. doi: 10.3934/math.2025167

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Abstract

The paper presents a new type of generalized Apostol-type Frobenius–Euler polynomials and numbers with specific order $ \kappa $ and level $ m $. We establish fundamental identities and properties using generating function techniques, such as summation formulas, differential and integral relations, and addition theorems. Additionally, we explore the connections between these polynomials and the Stirling numbers of the second kind, as well as other polynomial families. Lastly, we derive a differential equation and a recurrence relation for these new classes of polynomials. Finally, we show applications that can be obtained using these polynomials where the graphs of the zero functions and the meshes are displayed.

References

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