I am new to heat kernels for elliptic operators. When using the eigenfunction expansion method, I came across several expressions involving the Hermite polynomials $H_n$. After some routine computation towards the Green's function I arrived at the following integral:
$$\int_{0}^{+\infty}\left[\frac{1}{\sqrt{1-e^{-t}}}\exp\left(\frac{2xye^{-\frac{t}{2}}-\left(x^2+y^2\right)e^{-t}}{4\left(1-e^{-t}\right)}\right)-1\right]dt.\tag{$\star$}$$
My goal is to find the closed form of the above integral. I have noticed that by Mehler's formula, if we denote $\rho=e^{-t/2}$ then
\begin{align*} (\star)&=\int_{0}^{+\infty}\left[\frac{1}{\sqrt{1-\rho^2}}\exp\left(-\frac{\rho^2\left(\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2\right)-2\rho\left(\frac{x}{2}\right)\left(\frac{y}{2}\right)}{1-\rho^2}\right)-1\right]dt\\ &=\int_{0}^{+\infty}\sum_{n=1}^{\infty}\frac{\left(\frac{\rho}{2}\right)^n}{n!}H_n\left(\frac{x}{2}\right)H_n\left(\frac{y}{2}\right)dt\quad(\text{since}\;H_0\equiv1)\\ &=\sum_{n=1}^{\infty}\frac{1}{2^nn!}H_n\left(\frac{x}{2}\right)H_n\left(\frac{y}{2}\right)\underbrace{\int_{0}^{+\infty}e^{-\frac{nt}{2}}dt}_{=\frac{2}{n}}\\ &=\sum_{n=1}^{\infty}\frac{1}{2^{n-1}n\cdot n!}H_n\left(\frac{x}{2}\right)H_n\left(\frac{y}{2}\right). \end{align*}
I have tried introducing a new parameter and differentiating / integrating with respect to it and then applied again Mehler's formula (in the reverse direction to what I display above), but all these attempts seemed to make the expression even more complicated. The $n$ in the denominator is the tricky part.
Can anyone proceed from here or use other methods to evaluate ($\star$)? Any idea is much appreciated.
Note: I have posted this question on Mathematics Stack Exchange.
