Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 25, 2023

Umbral treatment and lacunary generating function for Hermite polynomials

  • Nusrat Raza and Umme Zainab EMAIL logo

Abstract

In this paper, we discuss umbral nature of 2-variable Hermite polynomials and obtain triple lacunary generating function for this polynomial by using the umbral method. Further, the lacunary generating function for 3-variable Hermite polynomials are also obtained.

MSC 2020: 05A40; 33C45

References

[1] P. Appell and J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques. Polynômes d’Hermite, Gauthier-Villars, Paris, 1926. Search in Google Scholar

[2] G. Dattoli, Generalized polynomials, operational identities and their applications. Higher transcendental functions and their applications, J. Comput. Apll. Math. 118 (2000), 111–123. 10.1016/S0377-0427(00)00283-1Search in Google Scholar

[3] G. Dattoli, B. Germano, S. Licciardi and M. R. Martinelli, On umbral treatment of Gegenbauer, Lagendre and Jacobi polynomialls, Int. Math. Forum. 12 (2017), no. 11, 531–551. 10.12988/imf.2017.6789Search in Google Scholar

[4] G. Dattoli, B. Germano, M. R. Martinelli and P. E. Ricci, Lacunary generating functions of Hermite polynomials and symbolic methods, Ilirias J. Math. 4 (2015), 16–23. Search in Google Scholar

[5] G. Dattoli, P. E. Ricci and C. Cesarano, A note on Legendre polynomials, Int. J. Nonlinear Sci. Numer. Simul. 2 (2001), no. 4, 365–370. 10.1515/IJNSNS.2001.2.4.365Search in Google Scholar

[6] G. Dattoli, P. E. Ricci and C. Cesarano, Monumbral polynomials and the associated formalism, Integral Transforms Spec. Funct. 13 (2002), no. 2, 155–162. 10.1080/10652460212901Search in Google Scholar

[7] G. Doetsch, Integraleigenschaften der Hermiteschen Polynome, Math. Z. 32 (1930), no. 1, 587–599. 10.1007/BF01194654Search in Google Scholar

[8] I. M. Gessel and P. Jayawant, A triple lacunary generating function for Hermite polynomials, Electron. J. Combin. 12 (2005), Research Paper 30. 10.37236/1927Search in Google Scholar

[9] N. Raza, U. Zainab, S. Araci and A. Esi, Identities involving 3-variable Hermite polynomials arising from umbral method, Adv. Difference Equ. 2020 (2020), Paper No. 640. 10.1186/s13662-020-03102-0Search in Google Scholar

[10] G.-C. Rota and B. D. Taylor, An introduction to the umbral calculus, Analysis, Geometry and Groups: A Riemann Legacy Volume, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor (1993), 513–525. Search in Google Scholar

[11] R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge University, Cambridge, 1999. 10.1017/CBO9780511609589Search in Google Scholar

[12] I. Vun and P. Belcher, Catalan numbers, Math. Spectrum 30 (1997), no. 1, 3–5. Search in Google Scholar

Received: 2023-02-16
Revised: 2023-03-24
Accepted: 2023-04-03
Published Online: 2023-07-25
Published in Print: 2023-12-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 26.10.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2023-2053/html
Scroll to top button