In a famous paper by Ait-Sahalia I have found this expression for the Hermite polynomial (pp 252, line -5):
$$ H_{j+1}^{\prime}\left(z\right)=-(1+j)\,H_j(z)\quad (1) $$
where $H_j$ is the $j$-th Hermite polynomial. The only relationship that links $H_{j+1}$ with $H_j$ that I know is
$$ H_{j+1}(z) = z\,H_j(z)-H_{j}^{\prime}(z)\quad (2) $$
which does not imply (1). More precisely taking the derivative of (2) I get
$$ H_{j+1}^{\prime}(z) = H_j(z)+z\,H^{\prime}_j(z)-H_{j}^{\prime\prime}(z). $$
So I am wondering where relationship (1) comes from.