I started with the following sums involving the product of Hermite polynomials.
$f(x,y;u)=\sum_{m=0}^{\infty}\frac{H_{2m}(x) H_{2m}(y)}{(2m)!}\big(\frac{u}{2}\big)^{2m} \tag{1}$
$g(x,y;u)=\sum_{m=0}^{\infty}\frac{H_{2m+1}(x) H_{2m+1}(y)}{(2m+1)!}\big(\frac{u}{2}\big)^{2m+1} \tag{2}$
where $u$ is a constant.
Using the relations of the Hermite polynomials, $H_n(x)$, to the Kummer's confluent hypergeometric function, $ {}_1 F_1(a;b;x)$, for even and odd $n$, I rewrite the sums $(1)$ and $(2)$ as follows.
$f(x,y;u)=\sum_{m=0}^{\infty}\frac{{}_1 F_1(-m;\frac{1}{2};x^2){}_1 F_1(-m;\frac{1}{2};y^2)}{m!} \big(\frac{u}{\sqrt{2}}\big)^{2m} \tag{3}$
$g(x,y;u)=\sum_{m=0}^{\infty}\frac{4xy{}_1 F_1(-m;\frac{3}{2};x^2){}_1 F_1(-m;\frac{3}{2};y^2)(2m+1)!}{(m!)^2} \big(\frac{u}{2}\big)^{2m+1} \tag{4}$
I'm wondering how to compute the sums $(3)$ and $(4)$ involving the product of hypergeometric functions. Any ideas on this would be much appreciated.
Addendum: I understand that the sum analogous to $(3)$ involving the Hermite polynomials i.e., $\sum_{m=0}^{\infty}\frac{H_{m}(x) H_{m}(y)}{m!}\big(\frac{u}{2}\big)^{m}$ has a closed-form expression given by the Mehler's formula. Therefore, in this context, is there also a closed-form expression for the sum $(3)$ and possibly, $(4)$?