In the book "Algebraic function fields and codes" by H. Stichtenoth, pg. 332 he treats cyclic-p Galois extensions $L/K$ in positive characteristic and says:
An element $\gamma_1\in L$ such that $L=K(\gamma_1)$ and $\gamma_1^p-\gamma_1\in K$, is called an Artin-Schreier generator for $L/K.$ Any two Artin-Schreier generators $\gamma$ and $\gamma_1$ of $L/K$ are related as follows: $\gamma_1=\mu \gamma+(b^p-b)$ with $0\neq \mu\in \mathbb{Z}/p\mathbb{Z}$ and $b\in K.$
I think what he meant to say is that the relation would be $\gamma_1=\mu \gamma+b$. Checking an old paper of Hasse at Crelle seems to say the same thing:
The relation that Stichtenoth writes is the one connecting the constant terms, i.e. if $\gamma^p-\gamma=b$ then the constant term of the generator given by $\gamma_1=\mu \gamma+b_1$ would be $\mu b +(b_1^p-b_1)$