This is a crosspost from Math Stack Exchange, please let me know if this is not an appropriate use of crossposting, and I will delete.
Mehler's formula is the following identity for Hermite polynomials $H_n(x)$:
$$\sum_{n=0}^{\infty}\frac{t^n}{2^nn!}H_n(x)H_n(y)=\frac{1}{\sqrt{1-t^2}}\exp\Bigg(-\frac{t^2(x^2+y^2)-2txy}{1-t^2}\Bigg)$$
for $|t|<1$. I'm curious if there is a similar formula for a finite sum $$\sum_{n=0}^{N}\frac{t^n}{2^nn!}H_n(x)H_n(y)$$
The Christoffel-Darboux formula offers an answer for the $t=1$ case $$\sum_{n=0}^{N}\frac{1}{2^nn!}H_n(x)H_n(y)=\frac{1}{N!2^{N+1}}\frac{H_N(y)H_{N+1}(x)-H_N(x)H_{N+1}(y)}{x-y}$$
Is there a more general formula for $t\neq 1$?
After playing around with it a bit, I've reduced the sum to a double integral:
$$\sum_{n=0}^{N}\frac{t^n}{2^nn!}H_n(x)H_n(y)=\frac{1}{4\pi}e^{x^2+y^2}\int_{-\infty}^{\infty}du\int_{-\infty}^{\infty}dv\frac{1}{N!}\Gamma(N+1,-\frac{uvt}{2})e^{-uvt/2}e^{iux+ivy-u^2/4-v^2/4}$$
Where $\Gamma(N+1,-\frac{uvt}{2})$ is the incomplete Gamma function $$\Gamma(N+1,-\frac{uvt}{2})=\int_{-\frac{uvt}{2}}^{\infty}dr\,r^Ne^{-r}=\Big(\frac{uvt}{2}\Big)^{N+1}\int_{-1}^{\infty}dr\,r^Ne^{-uvtr/2}$$
Though I am having a hard time finding a closed form expression for this integral.
