New Integral Equations for the Monic Hermite Polynomials


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Authors: K. ALI KHELIL, R. SFAXI AND A. BOUKHEMIS

DOI: 10.46793/KgJMat2201.007K

Abstract:



Keywords:

Linear functional, integral equation, integral representation on the real line, Hermite polynomials, Dawson function, Dirac mass.



References:

[1]    K.  Ali Khelil, R. Sfaxi and A.  Boukhemis, Integral representation of the generalized Bessel linear functional, Bull. Math. Anal. Appl. 9(3) (2017 ), 1–15.

[2]   N.  I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh, London, 1965.

[3]   T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

[4]   A.  Ghressi and L.  Kheriji, Some new results about a symmetric D-semiclassical linear form of class one, Taiwanese J. Math. 11(2) (2007 ), 371–382.

[5]   M.  E. H. Ismail and D.  Stanton, Classical orthogonal polynomials as moments, Canad. J. Math. 49 (1997  ), 520–542.

[6]   M.  E. H. Ismail and D.  Stanton, More orthogonal polynomials as moments, in: Mathematical Essays in Honor of Gian-Carlo Rota, Cambridge, MA, 1996, Birkhäuser Boston, Boston, MA, 1998, 377–396.

[7]   M.  E. H. Ismail and D.  Stanton, q-Integral and moment representations for q-orthogonal polynomials, Canad. J. Math. 45 (2002 ), 709–735.

[8]   N.  N. Lebedev, Special Functions and their Applications, Translated from the Russian by Richard A. Silverman, Englewood Cliffs, New Jork, 1965.

[9]   P.  Maroni, Une théorie algébrique des polynômes orthogonaux, Applications aux polynômes orthogonaux semi-classiques, IMACS: International Association for Mathematics and Computers in Simulation 9 (1991 ), 95–130.

[10]   P.  Maroni, Fonctions eulériennes, polynémes orthogonaux classiques, Techniques de l’Ingénieur 154 (1994  ), 1–30.

[11]   P. Maroni, An integral representation for the Bessel form, J. Comput. Appl. Math. 157 (1995 ), 251–260.

[12]   M.  Rahman and A.  Verma, A q-integral representation for the Rogers q-ultraspherical polynomials and some applications, Constr. Approx. 2 (1986  ), 1–10.

[13]   J.  Shohat and J.  D. Tamarkin, The Problem of Moments, American Mathematical Society, Providence, 1950.

[14]   B.  Simon, The classical moment as a selfadjoint finite difference operator, Adv. Math. 137 (1998  ), 82–203.

[15]   G.  Szegö, Orthogonal Polynomials, Fourth Edition, American Mathematical Society, Providence, 1975.