The DLMF has a hypergeometric representation for the Hermite polynomial $H_n(x)$ for real $x$, apparently.
This will give wrong results numerically with $n=333,x=1+i$ in Maxima and mpmath when compared with substitution into the polynomial. Is there such a representation for complex $z$? What would it be?
The motivation is to avoid substitution into the polynomial in the open source CAS Sage for precision reasons.