Operational methods for Hermite polynomials with applications

We exploit methods of operational nature to derive a set of new identities involving families of polynomials associated with operators providing different realizations of the Weyl group. The identities, we will deal with, extend the Nielsen formulae, valid for ordinary Hermite to families of Hermitelike polynomials. It will also be shown that the underlying formalism yields the possibility of obtaining further identities relevant to multi-variable and multi-index polynomials. Applications of higher order Hermite polynomials have been underlined for purpose of numerical simulation in continuous damage mechanics.

[1]  C. Cesarano,et al.  Bernstein polynomials and operational methods , 2006 .

[2]  J. L. Burchnall.  A NOTE ON THE POLYNOMIALS OF HERMITE , 1941 .

[3]  Ugo Andreaus,et al.  Soft-impact dynamics of deformable bodies , 2013 .

[4]  Bernardino Chiaia,et al.  Robustness-oriented design of a panel-based shelter system in critical sites , 2012 .

[5]  Jean-Jacques Marigo,et al.  The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models , 2011 .

[6]  P. E. Ricci,et al.  On a family of hybrid polynomials , 2004 .

[7]  Wilhelm Magnus,et al.  Lie Theory and Special Functions , 1969 .

[8]  Victor A. Eremeyev,et al.  Acceleration waves in micropolar elastic media , 2005 .

[9]  A. Misra,et al.  Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity , 2012 .

[10]  Luca Placidi,et al.  A variational approach for a nonlinear 1-dimensional second gradient continuum damage model , 2015 .

[11]  Francesco dell’Isola,et al.  Variational formulation of pre-stressed solid-fluid mixture theory, with an application to wave phenomena , 2008 .

[12]  C. Cesarano,et al.  On a new family of Hermite polynomials associated to parabolic cylinder functions , 2003, Appl. Math. Comput..

[13]  H. Altenbach,et al.  Acceleration waves and ellipticity in thermoelastic micropolar media , 2010 .

[14]  Dusan Krajcinovic,et al.  Statistical Damage Mechanics— Part I: Theory , 2005 .

[15]  Angelo Luongo,et al.  Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures , 2001 .

[16]  Clemente Cesarano.  Identities and generating functions on Chebyshev polynomials , 2012 .

[17]  Francesco dell’Isola,et al.  Modelling the onset of shear boundary layers in fibrous composite reinforcements by second gradient theory , 2013 .

[18]  Clemente Cesarano,et al.  A note on Generalized Bessel Functions , 2014 .

[20]  Preconditioning for solving Hermite Collocation by the Bi-CGSTAB , 2006 .

[21]  H. W. Gould,et al.  Operational formulas connected with two generalizations of Hermite polynomials , 1962 .

[22]  P. Appell,et al.  Fonctions hypergéométriques et hypersphériques : polynomes d'Hermite , 1926 .