Fractional Hermite-Kampé de Fériet and related polynomials. (English) Zbl 07915037
Summary: The goal of this paper is to demonstrate that incorporating the fractional form of the exponential function allows for an extension of special polynomials, such as the Hermite-Kampé de Fériet (H-KdF) polynomials with two variables, and the Mittag-Leffler-Gould-Hopper polynomials. This extension can be achieved by defining expansions using fractional powers, while still preserving the essential properties of the corresponding polynomial versions. Specifically, a fractional version of the classical Hermite polynomials is derived. The main properties are based on expansions in integer powers.
MSC:
| 33C70 | Other hypergeometric functions and integrals in several variables |
| 33C65 | Appell, Horn and Lauricella functions |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Samko, S., Kilbas, A.A., Marichev, O. Fractional Integrals and Derivatives, Taylor & Francis, Abingdon, Oxfordshire U.K., 1993. · Zbl 0818.26003 |
| [2] | Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and applications of fractional differential equations, 2006, North-Holland Math. Studies, vol. 204. · Zbl 1092.45003 |
| [3] | Gorenflo, F., Mainardi F. Fractional Calculus: Integral and Differential Equations of Fractional Order, in Fractals and Fractional Calculus in Continuum Mechanics, pp. 223-276, A. Carpinteri and F. Mainardi (Eds.), Springer, New York 1997. · Zbl 1438.26010 |
| [4] | Mainardi F., Luchko Y., Pagnini, G. The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 2001, 4, 153-192. · Zbl 1054.35156 |
| [5] | Beghin, L., Caputo, M. Commutative and associative properties of the Caputo fractional deriva-tive and its generalizing convolution operator, doi.org/10.1016/j.cnsns.2020.105338 · Zbl 1451.26007 |
| [6] | Groza, G., Jianu, M., Functions represented into fractional Taylor series, ITM Web of Con-ferences 29, 01017 (2019), (ICCMAE 2018); doi.org/10.1051/itmconf/20192901017. · doi:10.1051/itmconf/20192901017 |
| [7] | Caratelli, D., Natalini, P., Ricci, P.E., Fractional differential equations and expansions in fractional powers, Symmetry (2023), 15, 1842. https://doi.org/ 10.3390/sym15101842. · doi:10.3390/sym15101842 |
| [8] | Ricci, P.E., Natalini, P., Hypergeometric Bernoulli polynomials and r-associated Stirling num-bers of the second kind, Integers, 22 (2022), #A56, (12 pp). · Zbl 1498.33008 |
| [9] | Ricci, P.E., Srivastava, R., Natalini, P., A Family of the r-Associated Stirling Num-bers of the Second Kind and Generalized Bernoulli Polynomials, Axioms 2021, 10, 219. https://doi.org/10.3390/axioms10030219. · doi:10.3390/axioms10030219 |
| [10] | Khan, S., Wani, S.A., Riyasat, M., Study of generalized Legendre-Appell polynomials via frac-tional operators, TWMS Journal of Pure and Applied Mathematics, 11(2), 144-156 (2020). · Zbl 1517.33002 |
| [11] | Khan, S., Wani, S.A., Fractional calculus and generalized forms of special polynomials associ-ated with Appell sequences, Georgian Math. J., https://doi.org/10.1515/gmj-2019-2028. · Zbl 1464.26006 · doi:10.1515/gmj-2019-2028 |
| [12] | Goswami, A., Khan, M.F., Shadab, M., Study of Extended Hermite-Appell Polynomial via Fractional Operators, Appl. Math. Inf. Sci., 17(1), 27-33 (2023). |
| [13] | Wani, S.A., Riyasat, M., Integral transforms and extended Hermite-Apostol type Frobenius-Genocchi polynomials, Kragujevac J. Math., 48(1), 41-53 (2024) . |
| [14] | P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques. Polynômes d’Hermite, Gauthier-Villars, Paris, 1926. · JFM 52.0361.13 |
| [15] | Dattoli, G., Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle, in: Advanced Special Functions and Applications. Proc. Melfi School on Advanced Topics in Mathematics and Physics, Melfi, May 9-12, 1999 (D. Cocolicchio, G. Dattoli, and H.M. Srivastava, Eds.), Aracne Editrice, Rome (2000), pp. 147-164. · Zbl 1022.33006 |
| [16] | Dattoli, G., Ottaviani, P.L., Torre, A. and Vázquez L., Evolution operator equations: integra-tion with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cimento, 2, 1-133 (1997). |
| [17] | Ricci, P.E., Tavkhelidze, I., An introduction to operational techniques and special polynomials, Journal of Mathematical Sciences, Vol. 157, No. 1, 2009 · Zbl 1184.33001 |
| [18] | Widder, D.V., The Heat Equation, Academic Press, New York (1975). · Zbl 0322.35041 |
| [19] | Raza, N., Zainab, U., Mittag-Leffler-Gould-Hopper polynomials: Symbolic Approach, Rend. Circ. Mat. Palermo, Series 2, 2023. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
