Unraveling multivariable Hermite-Apostol-type Frobenius-Genocchi polynomials via fractional operators

Mohra Zayed; Shahid Ahmad Wani; Georgia Irina Oros; William Ramírez; Mohra Zayed; Shahid Ahmad Wani; Georgia Irina Oros; William Ramírez

doi:10.3934/math.2024840

AIMS Mathematics

2024, Volume 9, Issue 7: 17291-17304. doi: 10.3934/math.2024840

Previous Article Next Article

Research article

Unraveling multivariable Hermite-Apostol-type Frobenius-Genocchi polynomials via fractional operators

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India
3.
Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, Oradea 410087, Romania
4.
Section of Mathematics International Telematic University Uninettuno, Rome 00186, Italy
5.
Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla 080002, Colombia

Received: 05 March 2024 Revised: 22 April 2024 Accepted: 30 April 2024 Published: 20 May 2024
MSC : 11T23, 33B10, 33C45, 33E20, 33E30

This study explores the evolution and application of integral transformations, initially rooted in mathematical physics but now widely employed across diverse mathematical disciplines. Integral transformations offer a comprehensive framework comprising recurrence relations, generating expressions, operational formalism, and special functions, enabling the construction and analysis of specialized polynomials. Specifically, the research investigates a novel extended family of Frobenius-Genocchi polynomials of the Hermite-Apostol-type, incorporating multivariable variables defined through fractional operators. It introduces an operational rule for this generalized family, establishes a generating connection, and derives recurring relations. Moreover, the study highlights the practical applications of this generalized family, demonstrating its potential to provide solutions for specific scenarios.
- operational connection,
- fractional operators,
- Eulers' integral,
- multivariable special polynomials,
- explicit form,
- applications
Citation: Mohra Zayed, Shahid Ahmad Wani, Georgia Irina Oros, William Ramírez. Unraveling multivariable Hermite-Apostol-type Frobenius-Genocchi polynomials via fractional operators[J]. AIMS Mathematics, 2024, 9(7): 17291-17304. doi: 10.3934/math.2024840

Related Papers:

Abstract

This study explores the evolution and application of integral transformations, initially rooted in mathematical physics but now widely employed across diverse mathematical disciplines. Integral transformations offer a comprehensive framework comprising recurrence relations, generating expressions, operational formalism, and special functions, enabling the construction and analysis of specialized polynomials. Specifically, the research investigates a novel extended family of Frobenius-Genocchi polynomials of the Hermite-Apostol-type, incorporating multivariable variables defined through fractional operators. It introduces an operational rule for this generalized family, establishes a generating connection, and derives recurring relations. Moreover, the study highlights the practical applications of this generalized family, demonstrating its potential to provide solutions for specific scenarios.

References

[1]	G. Dattotli, S. Lorenzutta, C. Cesarano, Bernstein polynomials and operational methods, J. Comput. Anal. Appl., 8 (2006), 369–377.
[2]	G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle, In: Advanced special functions and applications, Rome: Aracne Editrice, 2000,147–164.
[3]	T. Nahid, J. Choi, Certain hybrid matrix polynomials related to the Laguerre-Sheffer family, Fractal Fract., 6 (2022), 211. https://doi.org/10.3390/fractalfract6040211 doi: 10.3390/fractalfract6040211
[4]	S. A. Wani, K. Abuasbeh, G. I. Oros, S. Trabelsi, Studies on special polynomials involving degenerate Appell polynomials and fractional derivative, Symmetry, 15 (2023), 840. https://doi.org/10.3390/sym15040840 doi: 10.3390/sym15040840
[5]	R. Alyusof, S. A. Wani, Certain properties and applications of $\Delta_h$ hybrid special polynomials associated with Appell sequences, Fractal Fract., 7 (2023), 233. https://doi.org/10.3390/fractalfract7030233 doi: 10.3390/fractalfract7030233
[6]	H. M. Srivastava, G. Yasmin, A. Muhyi, S. Araci, Certain results for the twice-iterated 2D $q$-Appell polynomials, Symmetry, 11 (2019), 1307. https://doi.org/10.3390/sym11101307 doi: 10.3390/sym11101307
[7]	A. M. Obad, A. Khan, K. S. Nisar, A. Morsy, q-Binomial convolution and transformations of $q$-Appell polynomials, Axioms, 10 (2021), 70. https://doi.org/10.3390/axioms10020070 doi: 10.3390/axioms10020070
[8]	D. Bedoya, O. Ortega, W. Ramírez, U. Urieles, New biparametric families of Apostol-Frobenius- Euler polynomials of level m, Mat. Stud., 55 (2021), 10–23. https://doi.org/10.30970/ms.55.1.10-23 doi: 10.30970/ms.55.1.10-23
[9]	G. Dattoli, Generalized polynomials operational identities and their applications, J. Comput. Appl. Math., 118 (2000), 111–123. https://doi.org/10.1016/S0377-0427(00)00283-1 doi: 10.1016/S0377-0427(00)00283-1
[10]	P. Appell, J. K. de Fériet, Fonctions hyperg${\acute{e}}$om${\acute{e}}$triques et hypersph${\acute{e}}$riques: polyn${\hat{o}}$mes d'Hermite, Paris: Gauthier-Villars, 1926.
[11]	L. C. Andrews, Special functions for engineers and applied mathematicians, New York: Macmillan Publishing Company, 1985.
[12]	G. Dattoli, Summation formulae of special functions and multivariable Hermite polynomials, Nuovo Cimento B, 119B (2004), 479–488. https://doi.org/10.1393/ncb/i2004-10111-1 doi: 10.1393/ncb/i2004-10111-1
[13]	M. A. Özarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 62 (2011), 2452–2462. https://doi.org/10.1016/j.camwa.2011.07.031 doi: 10.1016/j.camwa.2011.07.031
[14]	Q. M. Luo, Apostol-Euler polynomials of higher order and the Gaussian hypergeometric function, Taiwanese J. Math., 10 (2006), 917–925. https://doi.org/10.11650/twjm/1500403883 doi: 10.11650/twjm/1500403883
[15]	A. Erd${\rm\acute e}$lyi, Higher transcendental functions, McGraw-Hill Book Company, 1955.
[16]	L. Carlitz, Eulerian numbers and polynomials, Math. Mag., 32 (1959), 247–260. https://doi.org/10.2307/3029225 doi: 10.2307/3029225
[17]	K. B. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, New York: Academic Press, 1974.
[18]	D. V. Widder, An introduction to transform theory, New York: Academic Press, 1971.
[19]	G. Dattoli, P. E. Ricci, C. Cesarano, L. Vázquez, Special polynomials and fractional calculus, Math. Comput. Modell., 37 (2003), 729–733. https://doi.org/10.1016/S0895-7177(03)00080-3 doi: 10.1016/S0895-7177(03)00080-3
[20]	D. Assante, C. Cesarano, C. Fornaro, L. Vazquez, Higher order and fractional diffusive equations, J. Eng. Sci. Technol. Rev., 8 (2015), 202–204. https://doi.org/10.25103/JESTR.085.25 doi: 10.25103/JESTR.085.25
[21]	J. F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta. Math., 73 (1941), 333–366.
[22]	B. Kurt, Y. Simsek, Frobenius-Euler type polynomials related to Hermite-Bernoulli polyomials, AIP Conf. Proc., 1389 (2011), 385–388. https://doi.org/10.1063/1.3636743 doi: 10.1063/1.3636743
[23]	Y. Simsek, Generating functions for $q$-Apostol-type Frobenius-Euler numbers and polynomials, Axioms, 1 (2012), 395–403. https://doi.org/10.3390/axioms1030395 doi: 10.3390/axioms1030395
[24]	D. S. Kim, T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, J. Inequal. Appl., 307 (2012), 307. https://doi.org/10.1186/1029-242X-2012-307 doi: 10.1186/1029-242X-2012-307

Reader Comments

Your name:*

Email:*
© 2024 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)