On Mittag-Leffler-Gegenbauer polynomials arising by the convolution of Mittag-Leffler function and Hermite polynomials

Mohra Zayed; Maged G. Bin-Saad; Waleed K. Mohammed; Mohra Zayed; Maged G. Bin-Saad; Waleed K. Mohammed

doi:10.3934/math.2025746

AIMS Mathematics

2025, Volume 10, Issue 7: 16642-16663. doi: 10.3934/math.2025746

Previous Article Next Article

Research article Special Issues

On Mittag-Leffler-Gegenbauer polynomials arising by the convolution of Mittag-Leffler function and Hermite polynomials

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Department of Mathematics, College of Education, University of Aden, Yemen

Received: 28 April 2025 Revised: 26 June 2025 Accepted: 04 July 2025 Published: 24 July 2025
MSC : 33C45, 33E12, 26A33

Gegenbauer polynomials hold a significant role in the constructive theory of spherical functions, while the Mittag-Leffler function is widely used in fractional calculus. In this paper, we introduce a new class of Mittag-Leffler-Gegenbauer polynomials (MLGPs) by convolutionally combining the classical Hermite polynomials with the Mittag-Leffler function of three parameters. We explore some of its aspects, such as symbolical identities, recurrence relations, differential equations, generating functions, integral representations, finite summations, and Rodrigues-type and orthogonal formulas. Additionally, we demonstrate the relevance of the MLGPs by developing and solving a fractional kinetic equation associated with the MLGPs in the kernel. Finally, employing Saigo fractional-type operators, we establish fractional integrals and derivatives formulae for our innovative MLGPs. We conclude by proposing an open question regarding the Hermite numbers and their umbral calculus for further discussion in the field of this study.
- Mittag-Leffer function,
- Gegenbauer polynomials,
- Hermite polynomials,
- recurrence relations,
- integrals,
- fractional calculus,
- kinetic equation,
- orthogonality,
- symbolic operator
Citation: Mohra Zayed, Maged G. Bin-Saad, Waleed K. Mohammed. On Mittag-Leffler-Gegenbauer polynomials arising by the convolution of Mittag-Leffler function and Hermite polynomials[J]. AIMS Mathematics, 2025, 10(7): 16642-16663. doi: 10.3934/math.2025746

Related Papers:

Abstract

Gegenbauer polynomials hold a significant role in the constructive theory of spherical functions, while the Mittag-Leffler function is widely used in fractional calculus. In this paper, we introduce a new class of Mittag-Leffler-Gegenbauer polynomials (MLGPs) by convolutionally combining the classical Hermite polynomials with the Mittag-Leffler function of three parameters. We explore some of its aspects, such as symbolical identities, recurrence relations, differential equations, generating functions, integral representations, finite summations, and Rodrigues-type and orthogonal formulas. Additionally, we demonstrate the relevance of the MLGPs by developing and solving a fractional kinetic equation associated with the MLGPs in the kernel. Finally, employing Saigo fractional-type operators, we establish fractional integrals and derivatives formulae for our innovative MLGPs. We conclude by proposing an open question regarding the Hermite numbers and their umbral calculus for further discussion in the field of this study.

References

[1]	J. Avery, Gegenbauer polynomials, In: Hyperspherical harmonics, Reidel Texts in the Mathematical Sciences, Springer, Dordrecht, 5 (1989), 25–46. https://doi.org/10.1007/978-94-009-2323-2_3
[2]	D. Babusci, G. Dattoli, K. Górska, On Mittag-Leffler function and associated polynomials, arXiv, 2012. https://doi.org/10.48550/arXiv.1206.3495
[3]	D. Babusci, G. Dattoli, M. Del Franco, Lectures on mathematical methods for physics, Thecnical Report, 2010.
[4]	Y. Cao, Y. Kao, Z. Wang, X. Yang, J. H. Park, W. Xie, Sliding mode control for uncertain fractional-order reaction-diffusion memristor neural networks with time delays, Neural Networks, 178 (2024), 106402. https://doi.org/10.1016/j.neunet.2024.106402 doi: 10.1016/j.neunet.2024.106402
[5]	G. Dattoli, S. Khan, U. Ahmad, Hermite numbers and new families of polynomials, arXiv, 2025. https://doi.org/10.48550/arXiv.2503.14930
[6]	G. Dattoli, K. Górska, A. Horzela, S. Licciardi, R. M. Pidatella, Comments on the properties of Mittag-Leffler function, Eur. Phys. J. Spec. Top. , 226 (2017), 3427–3443. https://doi.org/10.1140/epjst/e2018-00073-1 doi: 10.1140/epjst/e2018-00073-1
[7]	G. Dattoli, A. Torre, M. Carpanese, Operational rules, and arbitrary order Hermite generating functions, J. Math. Anal. Appl. , 227 (1998), 98–111. https://doi.org/10.1006/jmaa.1998.6080 doi: 10.1006/jmaa.1998.6080
[8]	G. Dattoli, P. L. Ottaviani, A. Torre, L. Vázquez, Evolution operator equations: integration with algebraic and finitedifference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cim. , 20 (1997), 3–133. https://doi.org/10.1007/BF02907529 doi: 10.1007/BF02907529
[9]	G. Dattoli, S. Licciardi, Operational, umbral methods, Borel transform and negative derivative operator techniques, Integr. Transf. Spec. F. , 31 (2020), 192–220. https://doi.org/10.1080/10652469.2019.1684487 doi: 10.1080/10652469.2019.1684487
[10]	P. J. Davis, Interpolation and approximation, Dover publications, 1975.
[11]	A. Eŕdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, McGraw-Hill, 1955.
[12]	R. Gorenflo, R. Mainardi, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math. , 118 (2000), 283–299. https://doi.org/10.1016/S0377-0427(00)00294-6 doi: 10.1016/S0377-0427(00)00294-6
[13]	Y. Kao, C. Wang, H. Xia, Y. Cao, Projective synchronization for uncertain fractional reaction-diffusion systems via adaptive sliding mode control based on finite-time scheme, In: Analysis and control for fractional-order systems, Springer, 2024, 141–163. https://doi.org/10.1007/978-981-99-6054-5_8
[14]	Y. Kao, C. Wang, H. Xia, Y. Cao, Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control, In: Analysis and control for fractional-order systems, Springer, 2024, 121–140. https://doi.org/10.1007/978-981-99-6054-5_7
[15]	A. A. Kilbas, M. Saigo, On Mittag-Leffler type function, fractional calculas operators and solutions of integral equations, Integr. Transf. Spec. F. , 4 (1996), 355–370. https://doi.org/10.1080/10652469608819121 doi: 10.1080/10652469608819121
[16]	A. A. Kilbas, M. Saigo, R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr. Transf. Spec. F. , 15 (2004), 31–49. https://doi.org/10.1080/10652460310001600717 doi: 10.1080/10652460310001600717
[17]	J. D. E. Konhause, Bi-orthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. , 21 (1967), 303–314. https://doi.org/10.2140/pjm.1967.21.303 doi: 10.2140/pjm.1967.21.303
[18]	C. Kürt, M. A. Özarslan, Bivariate $k$-Mittag-Leffler functions with 2D-$k$-Laguerre-Konhauser polynomials and corresponding $k$-fractional operators, Miskolc Math. Notes, 24 (2023), 861–876. https://doi.org/10.18514/MMN.2023.3963 doi: 10.18514/MMN.2023.3963
[19]	A. M. Mathai, R. K. Saxena, H. J. Haubold, The $H$-function: theory and applications, Springer, 2010. https://doi.org/10.1007/978-1-4419-0916-9
[20]	G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha}(x)$, C. R. Acad. Sci. Paris, 137 (1903), 554–558.
[21]	M. A. Özarslan, C. Kürt, Bivariate Mittag-Leffler functions arising in the solutions of convolution integral equation with 2D-Laguerre-Konhauser polynomials in the kernel, Appl. Math. Comput. , 347 (2019), 631–644. https://doi.org/10.1016/j.amc.2018.11.010 doi: 10.1016/j.amc.2018.11.010
[22]	T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. , 19 (1971), 7–15.
[23]	I. Podlubny, Fractional differential equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, 1999.
[24]	E. D. Rainville, Special functions, New York: Macmillan, 1960.
[25]	N. Raza, U. Zainab, Mittag-Leffler-Gould-Hopper polynomials: symbolic approach, Rend. Circ. Mat. Palermo, II. Ser., 73 (2024), 1009–1036. https://doi.org/10.1007/s12215-023-00966-5 doi: 10.1007/s12215-023-00966-5
[26]	R. K. Saxena, S. L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput., 199 (2008), 504–511. https://doi.org/10.1016/j.amc.2007.10.005 doi: 10.1016/j.amc.2007.10.005
[27]	R. K. Saxena, A. M. Mathai, H. J. Haubold, On fractional kinetic equations, Astrophys. Space Sci., 282 (2002), 281–287. https://doi.org/10.1023/A:1021175108964 doi: 10.1023/A:1021175108964
[28]	M. J. S. Shahwan, M. G. Bin-Saad A. A. Al-Hashami, Some properties of bivariate Mittag-Leffler function, J. Anal., 31 (2023), 2063–2083. https://doi.org/10.1007/s41478-023-00551-0 doi: 10.1007/s41478-023-00551-0
[29]	M. J. S. Shahwan, M. G. Bin-Saad, On a class of Mittag-Leffler-Konhauser two-dimensional polynomials, Res. Math., 12 (2025), 2447647. https://doi.org/10.1080/27684830.2024.2447647 https://doi.org/10.1080/27684830.2024.2447647 doi: 10.1080/27684830.2024.2447647
[30]	M. S. Stankovć, S. D. Marinković, P. M. Rajković, The deformed and modified Mittag-Leffler polynomials, Math. Comput. Model., 54 (2011), 721–728. https://doi.org/10.1016/j.mcm.2011.03.016 doi: 10.1016/j.mcm.2011.03.016
[31]	H. M. Srivastava, H. L. Manocha, A treatise on generating functions, Halsted Press, 1984.
[32]	K. Subuhi, M. Ali, S. A. Naikoo, 2D-Sheffer-Mittag-Leffler polynomials: properties and examples, J. Classical Anal., 15 (2019), 99–117. https://doi.org/10.7153/jca-2019-15-10 doi: 10.7153/jca-2019-15-10
[33]	A. Wiman, Über den fundamentalsatz in der teorie der funktionen $E_{\alpha}(x)$, Acta Math., 29 (1905), 191–201. https://doi.org/10.1007/BF02403202 doi: 10.1007/BF02403202
[34]	P. Van Mieghem, The Mittag-Leffler function, arXiv, 2021. https://doi.org/10.48550/arXiv.2005.13330
[35]	E. M. Wright, The asymptotic expansion of the generalized Bessel functions, Proc. London Math. Soc., 38 (1934), 257–270. https://doi.org/10.1112/plms/s2-38.1.257 doi: 10.1112/plms/s2-38.1.257

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)