In forecasting principles and practice, the update equations are given as:
$$l_t = \alpha\frac{y_t}{s_{t-m}} + (1-\alpha)(l_{t-1} + b_{t-1})$$ $$b_t = \beta^*(l_t-l_{t-1}) + (1-\beta^*)b_{t-1}$$ $$s_t = \gamma\frac{y_t}{l_{t-1}+b_{t-1}} + (1-\gamma)s_{t-m}$$
However, in other texts I've found, such as this one and this one, the last one is given as
$$s_t = \gamma\frac{y_t}{l_t} + (1-\gamma)s_{t-m}$$
So, which is it?
As Hyndman writes regarding additive model, seasonal component there is usually expressed in one of two ways, either:
$$ s_{t}=\gamma\left (y_{t} - l_{t-1} - b_{t-1} \right ) + \left ( 1-\gamma \right )s_{t-m} $$
or:
$$ s_{t}=\gamma^{*}\left (y_{t} - l_{t} \right ) + \left ( 1-\gamma^{*} \right )s_{t-m}$$
where
$$ \gamma = \gamma^{*}\left (1-\alpha \right ) $$
Similarly for multiplicative model we have two representations:
$$ s_{t}=\gamma \frac{y_{t}}{l_{t-1} + b_{t-1}} + \left ( 1-\gamma \right )s_{t-m} $$
and
$$ s_{t}=\gamma^{*} \frac{y_{t}}{l_{t}} + \left ( 1-\gamma^{*} \right )s_{t-m} $$
Apparently Hyndman just didn't bother providing the second one.
