Mehler's Hermite Polynomial Formula
| sum_(n=0)^infty(H_n(x)H_n(y))/(n!)(1/2w)^n=(1-w^2)^(-1/2)exp[(2xyw-(x^2+y^2)w^2)/(1-w^2)], |
(1)
|
where H_n(x) is a Hermite polynomial (Watson 1933; Erdélyi 1938; Szegö 1975, p. 380). The generating function
| [画像: sum_(n=0)^infty(H_n(x))/(|_n/2_|!)w^n=(1+2xw+4w^2)/((1+4w^2)^(3/2))exp((4x^2w^2)/(1+4w^2)), ] |
(2)
|
where |_x_| is the floor function, can be derived from this equation (Doetsch 1930; Szegö 1975, p. 380). The more straightforward sum with |_n/2_| replaced by n in the denominator is given by
See also
Hermite PolynomialExplore with Wolfram|Alpha
WolframAlpha
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. https://www2.math.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.Referenced on Wolfram|Alpha
Mehler's Hermite Polynomial FormulaCite this as:
Weisstein, Eric W. "Mehler's Hermite Polynomial Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MehlersHermitePolynomialFormula.html