Abstract
This article deals with the introduction of Gould–Hopper based Frobenius–Genocchi polynomials and derivation of their properties. The summation formulae and operational rule for these polynomials are derived. In addition, the integral transforms and operational rules are used to obtain generalized form of Gould–Hopper based Frobenius–Genocchi polynomials.
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Appell, P., Kampé de Fériet, J.: Fonctions Hypergéométriques et Hypersphériques: Polyn\({\hat{o}}\)mes d’Hermite. Gauthier-Villars, Paris (1926)
Bretti, G., Cesarano, C., Ricci, P.E.: Laguerre-type exponentials and generalized Appell polynomials. Comput. Math. Appl. 48, 833–839 (2004)
Cesarano, C.: Generalized special functions in the description of fractional diffusive equations. Commun. Appl. Ind. Math. 10, 31–40 (2019)
Cesarano, C.: Multi-dimensional Chebyshev polynomials: a non-conventional approach. Commun. Appl. Ind. Math. 10, 1–19 (2019)
Cesarano, C.: Integral representations and new generating functions of Chebyshev polynomials. Hacet. J. Math. Stat. 44(3), 535–546 (2015)
Cesarano, C., Cennamo, G.M., Placidi, L.: Operational methods for Hermite polynomials with applications. Wseas Trans. Math. 13, 925–931 (2014)
Cesarano, C., Fornaro, C., Vazques, L.: Operational results on bi-orthogonal Hermite functions. Acta Math. Univ. Comenianae 85, 43–68 (2016)
Cesarano, C., Germano, B., Ricci, P.E.: Laguerre-type Bessel functions. Integr. Transforms Spec. Funct. 16(4), 315–322 (2005)
Costabile, F.A.: Modern Umbral Calculus, De Gruyter Studies in Mathematics, vol. 72, Berlin (2019)
Dattoli, G.: Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle. In: Advanced Special Functions and Applications (Melfi, 1999), Proc. Melfi Sch. Adv. Top. Math. Phys., vol. 1, pp. 147–164. Aracne, Rome (2000)
Dattoli, G.: Generalized polynomials, operational identities and their applications. J. Comput. Appl. Math. 118, 111–123 (2000)
Dattoli, G., Cesarano, C., Sacchetti, D.: A note on truncated polynomials. Appl. Math. Comput. 134(2–3), 595–605 (2003)
Dattoli, G., Lorenzutta, S., Cesarano, C.: Bernstein polynomials and operatinal methods. J. Comput. Anal. Appl. 8(4), 369–377 (2006)
Dattoli, G., Lorenzutta, S., Mancho, A.M., Torre, A.: Generalized polynomials and associated operational identities. J. Comput. Appl. Math. 108(1–2), 209–218 (1999)
Dattoli, G., Ottaviani, P.L., Torre, A., Vázquez, L.: Evolution operator equations: integration with algebraic and finite-difference methods. In: Applications to Physical Problems in Classical and Quantum Mechanics and Quantum Field Theory, Riv. Nuovo Cimento Soc. Ital. Fis., vol. 20, no. 2, pp. 1–133 (1997)
Dattoli, G., Ricci, P.E., Cesarano, C., Vázquez, L.: Special polynomials and fractional calculas. Math. Comput. Model. 37, 729–733 (2003)
Gould, H.W., Hopper, A.T.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29, 51–63 (1962)
Khan, S., Wani, S.A.: Extended Laguerre–Appell polynomials via fractional operators and their determinant forms. Turk J. Math. 42, 1686–1697 (2018)
Khan, S., Wani, S.A.: Fractional calculus and generalized forms of special polynomials associated with Appell sequences. Georgian Math. J. (2019)
Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, New York (1984)
Steffensen, J.F.: The poweriod, an extension of the mathematical notion of power. Acta Math. 73, 333–366 (1941)
Yaşar, B.Y., Özarslan, M.A.: Frobenius–Euler and Frobenius–Genocchi polynomials and their differential equations. New Trends Math. Sci. 3(2), 172–180 (2015)
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Wani, S.A., Khan, S. & Nahid, T. Gould–Hopper based Frobenius–Genocchi polynomials and their generalized form. Afr. Mat. 31, 1397–1408 (2020). https://doi.org/10.1007/s13370-020-00804-2
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DOI: https://doi.org/10.1007/s13370-020-00804-2