mehler's Hermite Polynomial Formula

(1)

where  is a Hermite polynomial (Watson 1933; Erdélyi 1938; Szegö 1975, p. 380). The generating function

(2)

where  is the floor function, can be derived from this equation (Doetsch 1930; Szegö 1975, p. 380). The more straightforward sum with  replaced by  in the denominator is given by

(3)

REFERENCES:

Almqvist, G. and Zeilberger, D. "The Method of Differentiating Under the Integral Sign." J. Symb. Comput. 10, 571-591, 1990.

Doetsch, G. "Integralgleichenschaften der Hermiteschen Polynome." Math. Z. 32, 587-599, 1930.

Erdélyi, A. "Über eine erzeugende Funktion von Produkten Hermitescher Polynome." Math. Z. 44, 201-211, 1938.

Foata, D. "A Combinatorial Proof of the Mehler Formula." J. Comb. Th. Ser. A 24, 250-259, 1978.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 194-195, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Rainville, E. D. Special Functions. New York: Chelsea, p. 198, 1971.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 380, 1975.

Watson, G. N. "Notes on Generating Functions of Polynomials: (2) Hermite Polynomials." J. London Math. Soc. 8, 194-199, 1933.