The first equation here is from Modern Control Engineering Katsuhiko Ogata. The second equation comes from an online tutorial here, equation 3 \begin{equation} G_c(s) = K_c\frac{s + \frac{1}{T}}{s + \frac{1}{\alpha T}}, \quad 0 < \alpha < 1 \quad \quad \quad (1)\end{equation}
and
\begin{equation} C(s) = a\frac{s + \frac{1}{aT}}{s + \frac{1}{T}}, \quad a > 1 \quad \quad \quad \quad \quad \quad(2)\end{equation}
Supposedly, these are equivalent lead compensators. But, I haven't had much luck proving they are equivalent, or turning one form into the other. Any help is appreciated, or a link to a reference that explains it is appreciated as well.
In both cases there two available variables involved in placing the pole and the zero. For the equations I assume $T$ can be chosen freely in either case. This means that for the first equation the constraints on $\alpha$ implies that pole should lie at a higher frequency than the zero. But is also means that for the second equation the constraint on $a$ implies that zero should lie at a lower frequency than the pole. So these constraints allows you to chose the available variables such that both equation can have the same pole and zero.