Hermite polynomials for fractional orders and negative arguments

Question

Hermite polynomials of order $n\in\mathbb{N}_0$ can be expressed as a special case of the confluent hypergeometric function (also called Tricomi's confluent hypergeometric function): $$H_n(x) = 2^n U\left(-\tfrac12 n, \tfrac12, x^2\right)$$ for $\operatorname{Re}(x)\geq0$. This representation allows us to generalize Hermite polynomials to negative and even fractional orders $n$ which is a great help in calculating some integrals. My problem is the following: I have Hermite polynomials of negative fractional order $-r$ and want to evaluate them at negative $x$, but the above representation does not exist, i.e. I cannot write $$H_{-r}(-x) = 2^{-r} U\left(\tfrac12 r, \tfrac12, x^2\right),$$ however using the known relation $H_n(-x)=(-1)^nH_n(x)$ also won't work as I get complex numbers then (because of $(-1)^{-r}$). Do representations of Hermite polynomials using hypergeometric and other functions exist that extend the region of convergence to points on the left half-plane $\operatorname{Re}(x)\leq0$? I couldn't find any.