Identities involving 3-variable Hermite polynomials arising from umbral method
Tóm tắt
In this paper, we employ an umbral method to reformulate the 3-variable Hermite polynomials and introduce the 4-parameter 3-variable Hermite polynomials. We also obtain some new properties for these polynomials. Moreover, some special cases are discussed and some concluding remarks are also given.
Tài liệu tham khảo
Andrews, L.C.: Special Functions for Applied Mathematics and Engineering. MacMillan, New York (1985)
Appell, P., Kampé de Fériet, J.: Fonctions Hypergéométriques et Hypersphériques: Polynômes d’ Hermite. Gauthier-Villars, Paris (1926)
Araci, S.: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl. Math. Comput. 233, 599–607 (2014)
Babusci, D., Dattoli, G., Del Franco, M.: Lectures on mathematical methods for physics. Internal. Report ENEA RT/2010/5837
Babusci, D., Dattoli, G., Quattromini, M.: On integrals involving Hermite polynomials. Appl. Math. Lett. 25(8), 1157–1160 (2012)
Dattoli, G.: Generalized polynomials, operational identities and their applications. Higher transcendental functions and their applications. J. Comput. Appl. Math. 118(1–2), 111–123 (2000)
Dattoli, G., Germano, B., Licciardi, S., Martinelli, M.R.: On umbral treatment of Gegenbauer, Legendre and Jacobi polynomials. Int. Math. Forum 12(11), 531–551 (2017)
Dattoli, G., Germano, B., Martinelli, M.R., Ricci, P.E.: Lacunary generating functions of Hermite polynomials and symbolic methods. Ilirias J. Math. 4, 16–23 (2015)
Dattoli, G., Ottaviani, P.L., Torre, A., Vázquez, L.: Evolution operator equations: integration with algebraic and finite-difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory. Riv. Nuovo Cimento Soc. Ital. Fis. (4) 20(2) (1997) 133 pp.
Dattoli, G., Ricci, P.E., Cesarano, C.: Monumbral polynomials and the associated formalism. Integral Transforms Spec. Funct. 13(2), 155–162 (2002)
Dattoli, G., Ricci, P.E., Khomasuridze, I.: On the derivation of new families of generating functions involving ordinary Bessel functions and Bessel-Hermite functions. Math. Comput. Model. 46(3–4), 410–414 (2007)
Gessel, I.M., Jayawant, P.: A triple lacunary generating function for Hermite polynomials. Electron. J. Combin. 12 (2005) Research Paper, 30, 14 pp.
Gould, H.W., Hopper, A.T.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29, 51–63 (1962)
Jang, L.-C., Kim, T., Kim, D.S., Kim, H.Y.: Extended r-central Bell polynomials with umbral calculus viewpoint. Adv. Differ. Equ. 2019, Article ID 202 (2019)
Kim, T., Kim, D.S.: Some identities of extended degenerate r-central Bell polynomials arising from umbral calculus. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114(1), Paper No. 1 (2020)
Kim, T., Kim, D.S., Kwon, H.-I., Kwon, J.: Umbral calculus approach to r Stirling number of the second kind and r Bell polynomials. J. Comput. Anal. Appl. 27, 173–188 (2019)
Lebedev, N.N.: Special Functions and Their Applications, Revised edn. Dover, New York (1972) xii+308 pp. Translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication
Li, T., Viglialoro, G.: Analysis and explicit solvability of degenerate tensorial problems. Bound. Value Probl. 2018, Article ID 2 (2018)
Louisell, W.H.: Quantum Statistical Properties of Radiation. Wiley Classics Library. A Wiley-Interscience Publication, Wiley, New York (1990) xvi+528 pp. Reprint of the 1973 edition
Viglialoro, G., Woolley, T.E.: Boundedness in a parabolic-elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source. Math. Methods Appl. Sci. 41(5), 1809–1824 (2018)
Widder, D.V.: The Heat Equation. Pure and Applied Mathematics, vol. 67, xiv+267 pp. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1975)