From the generating formula for Hermite polynomials we know that
$$ e^{2xz - z^2} = \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n!} \, . $$
The sum
$$ \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n! \sqrt{n+\frac{1}{2}}} $$
has appeared in some calculations and I can't really figure out how to solve it. I'm pretty sure it converges, since the denominator increases faster than in the generating formula expression. It might be necessary to expand the square root and truncate in first order in $n$, but even this approximation proved to be more complicated to solve than I thought, and I couldn't do it.
Edit: If it helps, this formula was extracted from the sum
$$ \sum_{n=0}^\infty \left( \frac{H_n(x) \, z^n}{n!} \right) \exp \left[ -it \left(n+\frac{1}{2} \right)^{\frac{1}{2}} \right] \, ,$$
which can maybe be easier to solve.