In [the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737][1], they attribute on page 731 the following formula to Mehler (quoting a book by Erdelyi):
$$
\sum_{n=0}^{\infty}\frac{r^n H_n(x) H_n(y)}{2^n n!}=(1-r^2)^{-1/2}
\exp\bigl\{
x^2-\frac{(x-ry)^2}{(1-r^2)}
\bigr\},
$$
where $H_n$ is the $n$th Hermite polynomial.

Question: I do not believe that formula, since the lhs is symmetric in $x,y$ whereas the rhs fails to be symmetric in $x,y$. I am also puzzled since, as said above, this article is a reference material for numerous articles on Laguerre and Hermite polynomials.


  [1]: https://doi.org/10.2307/2373813