Functional Analysis/Harmonic Analysis
A composition formula for squares of Hermite polynomials and its generalizations
Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 849-852.

We prove a general formula which, with appropriately chosen parameters, gives a composition formula for squares of Gould–Hopper polynomials g2n(x,h), and hence also for Hermite polynomials. Our main tool is the classical Mehler formula, but with imaginary arguments.

Nous démontrons une formule générale qui, avec des coefficients convenablement choisis, donne une formule de composition pour les carrés des polynômes de Gould–Hopper gn2(x,h) et, par conséquent, pour les carrés des polynômes d'Hermite. Notre outil principal est la formule de Mehler classique avec l'argument imaginaire.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.12.024
Graczyk, Piotr 1; Nowak, Adam 2

1 Département de mathématiques, Université d'Angers, 2, boulevard Lavoisier, 49045 Angers cedex 01, France
2 Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
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Graczyk, Piotr; Nowak, Adam. A composition formula for squares of Hermite polynomials and its generalizations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 849-852. doi : 10.1016/j.crma.2003.12.024. http://www.numdam.org/articles/10.1016/j.crma.2003.12.024/

[1] Gould, H.W.; Hopper, A.T. Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., Volume 29 (1962), pp. 51-63

[2] P. Graczyk, J.J. Loeb, I. Lopez, A. Nowak, W. Urbina, Sobolev spaces and fractional derivation for Laguerre expansions, 2003, submitted for publication

[3] Sjögren, P. Operators associated with the Hermite semigroup – a survey, J. Fourier Anal. Appl., Volume 3 (1997), pp. 813-823

[4] Srivastava, H.M.; Manocha, H.L. A Treatise on Generating Functions, Halsted, Wiley, New York, 1984

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