Mehler's Hermite Polynomial Formula

$$\sum_{n=0}^\infty {H_n(x)H_n(y)\over n!} ({\textstyle{1\over... ...\rm exp}\nolimits \left[{2xyw-(x^2+y^2)w^2\over 1-w^2}\right],$$

where $H_n(x)$ is a Hermite Polynomial.

References

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

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

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 194-195, 1996.

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.