A new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type

Ugur Duran; Can Kızılateş; William Ramírez; Clemente Cesarano; Ugur Duran; Can Kızılateş; William Ramírez; Clemente Cesarano

doi:10.3934/math.2025722

AIMS Mathematics

2025, Volume 10, Issue 7: 16117-16138. doi: 10.3934/math.2025722

Previous Article Next Article

Research article Special Issues

A new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type

1.
Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200, Hatay, Turkey
2.
Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, Zonguldak 67100, Turkey
3.
Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58, 55-66, 080002 Barranquilla, Colombia
4.
Section of Mathematics International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy

Received: 06 May 2025 Revised: 26 June 2025 Accepted: 03 July 2025 Published: 17 July 2025
MSC : 11B68, 33C45, 11B73, 05A19

In this paper, we consider a new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type, denoted by $ \mathcal{W}^{(\alpha)}_{n, \lambda}(\delta, \zeta; \rho; \mu) $. We obtain several summation formulae, a recurrence relation, two difference operator formulas, two derivative operator formulas, an implicit summation formula, and a symmetric property for these polynomials. Also, we provide a representation of the degenerate differential operator on the degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. Moreover, we define the degenerate unified Hermite-based Apostol-Stirling polynomials of the second kind and derive some properties of these newly established polynomials. In addition, we prove multifarious correlations, including the new polynomials. Furthermore, we list the first few degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type for some special cases and present data visualizations of zeros forming 2D and 3D structures. Finally, we provide a table covering approximate solutions for the zeros of $ \mathcal{W}^{(\alpha)}_{n, 3}(\delta, 4;3;2) $.
- Appell-type polynomials,
- Bernoulli numbers and polynomials,
- Euler numbers and polynomials,
- Hermite polynomials,
- unified polynomials,
- degenerate exponential function
Citation: Ugur Duran, Can Kızılateş, William Ramírez, Clemente Cesarano. A new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type[J]. AIMS Mathematics, 2025, 10(7): 16117-16138. doi: 10.3934/math.2025722

Related Papers:

Abstract

In this paper, we consider a new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type, denoted by $ \mathcal{W}^{(\alpha)}_{n, \lambda}(\delta, \zeta; \rho; \mu) $. We obtain several summation formulae, a recurrence relation, two difference operator formulas, two derivative operator formulas, an implicit summation formula, and a symmetric property for these polynomials. Also, we provide a representation of the degenerate differential operator on the degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. Moreover, we define the degenerate unified Hermite-based Apostol-Stirling polynomials of the second kind and derive some properties of these newly established polynomials. In addition, we prove multifarious correlations, including the new polynomials. Furthermore, we list the first few degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type for some special cases and present data visualizations of zeros forming 2D and 3D structures. Finally, we provide a table covering approximate solutions for the zeros of $ \mathcal{W}^{(\alpha)}_{n, 3}(\delta, 4;3;2) $.

References

[1]	N. G. Acala, A unification of the generalized multiparameter Apostol-type Bernoulli, Euler, Fubini, and Genocchi polynomials of higher order, Eur. J. Pure Appl. Math., 13 (2020), 587–607. https://doi.org/10.29020/nybg.ejpam.v13i3.3757 doi: 10.29020/nybg.ejpam.v13i3.3757
[2]	L. C. Andrews, Special functions of mathematics for engineers, 2 Eds., Spie Press, 1997. https://doi.org/10.1117/3.270709
[3]	P. Appell, J. K. de Fériet, Fonctions hypergéométriques et hypersphériques: polynomes d'Hermite, Paris: Gauthier-Villars, 1926.
[4]	S. Araci, Construction of degenerate $q$-Daehee polynomials with weight $\alpha $ and its applications, Funda. J. Math. Appl., 4 (2021), 25–32. https://doi.org/10.33401/fujma.837479 doi: 10.33401/fujma.837479
[5]	D. Bedoya, C. Cesarano, W. Ramírez, L. Castilla, A new class of degenerate biparametric Apostol-type polynomials, Dolomites Res. Notes Approx., 16 (2023), 10–19. https://doi.org/10.14658/PUPJ-DRNA-2023-1-2 doi: 10.14658/PUPJ-DRNA-2023-1-2
[6]	D. Bedoya, C. Cesarano, S. Díaz, W. Ramírez, New classes of degenerate unified polynomials, Axioms, 12 (2023), 21. https://doi.org/10.3390/axioms12010021 doi: 10.3390/axioms12010021
[7]	H. Belbachir, Y. Djemmada, S. Hadj-Brahim, Unified Bernoulli-Euler polynomials of Apostol type, Indian J. Pure Appl. Math., 54 (2023), 76–83. https://doi.org/10.1007/s13226-022-00232-x doi: 10.1007/s13226-022-00232-x
[8]	L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math., 15 (1979), 51–88.
[9]	L. Carlitz, A degenerate Staud-Clausen theorem, Arch. Math., 7 (1956), 28–33. https://doi.org/10.1007/BF01900520 doi: 10.1007/BF01900520
[10]	S. Diaz, W. Ramirez, C. Cesarano, J. Hernandez, E. C. P. Rodriguez, The monomiality principle applied to extensions of Apostol-type Hermite polynomials, Eur. J. Pure Appl. Math., 18 (2025), 5656. https://doi.org/10.29020/nybg.ejpam.v18i1.5656 doi: 10.29020/nybg.ejpam.v18i1.5656
[11]	U. Duran, P. N. Sadjang, On Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a $q$-parameter, Mathematics, 7 (2019), 121. https://doi.org/10.3390/math7020121 doi: 10.3390/math7020121
[12]	H. W. Gould, A. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29 (1962), 51–63. https://doi.org/10.1215/S0012-7094-62-02907-1 doi: 10.1215/S0012-7094-62-02907-1
[13]	T. Kim, A note on degenerate Stirling polynomials of the second kind, Proc. Jangjeon Math. Soc., 20 (2017), 319 –331.
[14]	T. Kim, D. S. Kim, On some degenerate differential and degenerate difference operators, Russ. J. Math. Phys., 29 (2022), 37–46. https://doi.org/10.1134/S1061920822010046 doi: 10.1134/S1061920822010046
[15]	T. Kim, D. S. Kim, J. Kwon, Probabilistic identities involving fully degenerate Bernoulli polynomials and degenerate Euler polynomials, Appl. Math. Sci. Eng., 33 (2025), 2448193. https://doi.org/10.1080/27690911.2024.2448193 doi: 10.1080/27690911.2024.2448193
[16]	T. Kim, D. S. Kim, W. Kim, J. Kwon, Some identities related to degenerate Bernoulli and degenerate Euler polynomials, Math. Comput. Model. Dyn. Syst., 30 (2024), 882–897. https://doi.org/10.1080/13873954.2024.2425155 doi: 10.1080/13873954.2024.2425155
[17]	B. Kurt, Unified degenerate Apostol-type Bernoulli, Euler, Genocchi, and Fubini polynomials, J. Math. Comput. Sci., 25 (2022), 259–268. https://doi.org/10.22436/jmcs.025.03.05 doi: 10.22436/jmcs.025.03.05
[18]	M. A. Özarslan, Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials, Adv. Differ. Equ., 2013 (2013), 116. https://doi.org/10.1186/1687-1847-2013-116 doi: 10.1186/1687-1847-2013-116
[19]	J. W. Park S. S. Pyo, A note on degenerate Bernoulli polynomials arising from umbral calculus, Adv. Stud. Contemp. Math., 32 (2022), 509–525. https://doi.org/10.17777/ascm2022.32.4.509 doi: 10.17777/ascm2022.32.4.509
[20]	M. A. Pathan, W. A. Khan, Unified Apostol type Hermite-Bernoulli, Hermite-Euler and Hermite-Genocchi polynomials, Sarajevo J. Math., 16 (2022), 163–178. https://doi.org/10.5644/SJM.16.02.03 doi: 10.5644/SJM.16.02.03
[21]	W. Ramírez, C. Kızı lateş, D. Bedoya, C. Cesarano, C. S. Ryoo, On certain properties of three parametric kinds of Apostol-type unified Bernoulli-Euler polynomials, AIMS Math., 10 (2025), 137–158. https://doi.org/10.3934/math.2025008 doi: 10.3934/math.2025008
[22]	H. M. Srivastava, B. Kurt, V. Kurt, Identities and relations involving the modified degenerate Hermite-based Apostol-Bernoulli and Apostol-Euler polynomials, RACSAM, 113 (2019), 1299–1313. https://doi.org/10.1007/s13398-018-0549-1 doi: 10.1007/s13398-018-0549-1
[23]	D. V. Widder, The heat equation, Academic Press: New York, NY, USA, 1975.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)