Mehler's Hermite Polynomial Formula for $|w|=1$ and applying the Dominated Convergence theorem to it

Question

I only know proofs of the Mehler formula see Wolfram Mathworld for the case $|w|<1$. In this paper (bottom of page 3), however, Mehler's formula is used for $|w|=\exp(-i\alpha)$. Moreover, they immediately proceed to exchange the integral with the limit of the series in Mehler's formula. If someone could explain both these steps to me, I'd be highly grateful, as I wish to do just these steps for my work, too.
In particular, I'd like to justify this equation's first and last step, i.e. (a) Can I exchange integral and limit and (b) can I apply Mehler's formula: $$\frac{d}{dx}\sum_{n=0}^\infty\left(\frac{w}{2}\right)^{2n+1}\frac{H_{2n+1}(x)H_{2n}(0)}{(2n+1)!}=2\sum_{n=0}^{\infty}\left(\frac{w}{2}\right)^{2n+1}\frac{H_{2n}(x)H_{2n}(0)}{(2n)!}\\ =w\sum_{n=0}^{\infty}\left(\frac{w}{2}\right)^{n}\frac{H_{n}(x)H_{n}(0)}{n!}\\ =w\cdot(1-w^2)^{-\frac 1 2}e^{-x^2\frac{w^2}{1-w^2}}$$ for $$w=e^{ia}, a\in \mathbb{R}$$ And thus (?) $$\sum_{n=0}^\infty\left(\frac{w}{2}\right)^{2n+1}\frac{H_{2n+1}(x)H_{2n}(0)}{(2n+1)!}=\frac{\sqrt[]{\pi}}{2}\mathrm{erf}(\frac{wx}{\sqrt{1-w^2}})$$ I tested it numerically for some values of $x$ and it seems to hold.