I'm trying to find a closed form of an infinite sum of a product of (physicist's) Hermite polynomials:
$S_1(x,y):=\sum_{k=1}^\infty \frac{1}{2^k k! k}H_k(x)H_k(y)$
This sum is very close to what appears to be an orthogonality condition as given in this page. This says that
$S_2(x,y):=\sum_{k=0}^\infty \frac{1}{2^k k!}H_k(x)H_k(y)=\sqrt{\pi}e^{\frac{1}{2}(x^2+y^2)}\delta(x-y)$
The difference between the sums is obviously the extra $1/k$ in the coefficient in $S_1$.
An attempt I performed was to use another result from the same link:
$S_3(x,y):=\sum_{k=0}^\infty \frac{1}{k!}H_k(x)H_k(y)w^k=\frac{1}{\sqrt{1-4w^2}}e^{\frac{2w(2w(x^2+y^2)-2xy)}{4w^2-1}}$
where I first subtract out the $k=0$ term, then divide by $w$ and then integrate to obtain the $1/k$ (setting $w=1/2$ eventually). However, the integral on the right-hand side does not appear to be easily computable.
Can a closed form expression be found for $S_1(x,y)$?