Operational results in bi-orthogonal Hermite functions
Main Article Content
Abstract
By starting from the concept of the orthogonality property related to the ordinary and generalized two-variable Hermite polynomials, we present some interesting results on the class of bi-orthogonal Hermite functions.The structure of these bi-orthogonal functions is based on the family of the two-index, two-variable Hermite polynomials of type $H_{m,n}(x; y)$ and their adjoint $G_{m,n}(x; y)$.We will also introduce a dierential representation of the operators acting on the above bi-orthogonal Hermite functions and we will derive some operational identities.
Article Details
How to Cite
Cesarano, C., Fornaro, C., & Vazquez, L.
(2016).
Operational results in bi-orthogonal Hermite functions.
Acta Mathematica Universitatis Comenianae, 85(1), 43-68.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/119/287
Issue
Section
Articles
References
1. P. Appell and J. Kampe de Feriet, Fonctions hypergeometriques et hyperspheriques. Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
2. C. Cesarano, G.M. Cennamo, L. Placidi, Humbert Polynomials and Functions in Terms of Hermite Polynomials towards Applications to Wave Propagation, WSEAS Transactions on Mathematics, 13 (2014), 595-602.
3. H.M. Srivastava, H.L. Manocha, A treatise on generating functions, Wiley, New York, 1984.
4. T.H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators I{II, Kon. Ned. Akad. Wet. Ser. A, 77, 46-66.
5. G. Dattoli and C. Cesarano, On a new family of Hermite polynomials associated with of parabolic cylinder functions, Applied Mathematics and Computation, Vol. 141 (1), 2003, 143-149
6. H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite Polynomials, Duke Math. J., 29 (1962), 51-62.
7. G. Dattoli, S. Lorenzutta, C. Cesarano and P.E. Ricci, Second level exponentials and families of Appell polynomials, Int. Transf. Spec. Funct., 13 (6) (2002), 521-527.
8. C. Cesarano, A Note on generalized Hermite Polynomials, Inter. Journal of Applied Mathematics and Informatics, 8 (2014), 1-6.
9. G. Dattoli, S. Lorenzutta, P.E. Ricci, C. Cesarano, On a family of hybrid polynomials, Int. Transf. Spec. Funct., 15 (6) (2004), 485-490.
10. G. Dattoli, C. Cesarano, D. Sacchetti, A note on truncated polynomials, Applied Mathematics and Computation, Vol. 134 (2-3) (2002), 595-605.
11. Y.S. Kim, M.E. Noz, Phase-space picture of quantum mechanics, World Scientific, Singapore, 1991.
12. H. Koch, D. Tataru, Lp Eigenfunctions bounds for the Hermite operator, Duke Math. J., 128 (2) 2005, 369-392.
13. C. Cesarano, D. Assante, A note on Generalized Bessel Functions, International Journal of Mathematical Models and Methods in Applied Sciences, 7 (6) (2014), 625-629.
14. D. Assante, D. Davino, S. Falco, F. Schettino, L. Verolino, Coupling impedance of a charge traveling in a drift tube, IEEE Transactions on Magnetics, 41 (5) (2005), 1924--1927.
15. D. Assante, L. Verolino, Efficient evaluation of the longitudinal coupling impedance of a plane strip, Progress In Electromagnetics Research M 26 (2012), 251-265.
16. D. Assante, C. Cesarano, Simple Semi-Analytical Expression of the Lightning Base Current in the Frequency-Domain, Journal of Engineering Science and Technology Review, 7 (2) (2014), 1-6.
17. D. Assante, C. Fornaro, Voltage calculation in periodically grounded multiconductor transmission lines, International Journal of Circuits, Systems and Signal Processing, 8 (2014), 54-60.
2. C. Cesarano, G.M. Cennamo, L. Placidi, Humbert Polynomials and Functions in Terms of Hermite Polynomials towards Applications to Wave Propagation, WSEAS Transactions on Mathematics, 13 (2014), 595-602.
3. H.M. Srivastava, H.L. Manocha, A treatise on generating functions, Wiley, New York, 1984.
4. T.H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators I{II, Kon. Ned. Akad. Wet. Ser. A, 77, 46-66.
5. G. Dattoli and C. Cesarano, On a new family of Hermite polynomials associated with of parabolic cylinder functions, Applied Mathematics and Computation, Vol. 141 (1), 2003, 143-149
6. H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite Polynomials, Duke Math. J., 29 (1962), 51-62.
7. G. Dattoli, S. Lorenzutta, C. Cesarano and P.E. Ricci, Second level exponentials and families of Appell polynomials, Int. Transf. Spec. Funct., 13 (6) (2002), 521-527.
8. C. Cesarano, A Note on generalized Hermite Polynomials, Inter. Journal of Applied Mathematics and Informatics, 8 (2014), 1-6.
9. G. Dattoli, S. Lorenzutta, P.E. Ricci, C. Cesarano, On a family of hybrid polynomials, Int. Transf. Spec. Funct., 15 (6) (2004), 485-490.
10. G. Dattoli, C. Cesarano, D. Sacchetti, A note on truncated polynomials, Applied Mathematics and Computation, Vol. 134 (2-3) (2002), 595-605.
11. Y.S. Kim, M.E. Noz, Phase-space picture of quantum mechanics, World Scientific, Singapore, 1991.
12. H. Koch, D. Tataru, Lp Eigenfunctions bounds for the Hermite operator, Duke Math. J., 128 (2) 2005, 369-392.
13. C. Cesarano, D. Assante, A note on Generalized Bessel Functions, International Journal of Mathematical Models and Methods in Applied Sciences, 7 (6) (2014), 625-629.
14. D. Assante, D. Davino, S. Falco, F. Schettino, L. Verolino, Coupling impedance of a charge traveling in a drift tube, IEEE Transactions on Magnetics, 41 (5) (2005), 1924--1927.
15. D. Assante, L. Verolino, Efficient evaluation of the longitudinal coupling impedance of a plane strip, Progress In Electromagnetics Research M 26 (2012), 251-265.
16. D. Assante, C. Cesarano, Simple Semi-Analytical Expression of the Lightning Base Current in the Frequency-Domain, Journal of Engineering Science and Technology Review, 7 (2) (2014), 1-6.
17. D. Assante, C. Fornaro, Voltage calculation in periodically grounded multiconductor transmission lines, International Journal of Circuits, Systems and Signal Processing, 8 (2014), 54-60.