I came across an infinite summation of the following form involving the product of two Hermite polynomials while solving a physics problem.
$f(x,y;u,a,b)=\sum_{n=0}^{\infty} \frac{(a+bn)H_{n}(x)H_{n}(y)}{n!}\big(\frac{u}{2}\big)^n$
where, $a$, $b$, and $u$ are constants.
I figured that one part of this sum i.e., $\sum_{n=0}^{\infty} \frac{a H_{n}(x)H_{n}(y)}{n!}\big(\frac{u}{2}\big)^n$ can be evaluated using the Mehler's formula but I'm wondering how to compute the other part which involves a factor of $n$ inside the sum i.e., $\sum_{n=0}^{\infty} \frac{b n H_{n}(x)H_{n}(y)}{n!}\big(\frac{u}{2}\big)^n$.
I tried partially differentiating the Mehler's fromula with respect to either $x$ (or $y$). However, I'm left with the sum: $\sum_{n=0}^{\infty}\frac{2n H_{n-1}(x)H_{n}(y)}{n!}\big(\frac{u}{2}\big)^n$ which is not quite what I want to compute the other part in my summation.
I'm looking to get a closed-form expression for the full summation if it converges. Any help on this would be much appreciated.