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From Goldstein we have that, for non-extended ($\lambda =1 $), the generating function of third type is $$F = F_3(p,Q,t) + q_ip_i.\tag{1}$$ Although I found it hard to see if that would hold true also for extended canonical transformations ($\lambda \neq 1$).
The only reference I found claims that in these cases $$F = F_3(p,Q,t) + \lambda q_ip_i\tag{2}$$ although they provide no proof nor explanation, and I have been having hard times convincing myself of that.

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OP's eq. (2) is correct. Using the definition of an extended CT: $$ \lambda(\sum_{i=1}^np_i\mathrm{d}q^i-H\mathrm{d}t) -(\sum_{i=1}^nP_i\mathrm{d}Q^i -K\mathrm{d}t) ~=~\mathrm{d}F,\tag{9.8}$$ we derive the extended type 3 conditions: $$ \begin{align} \lambda q^i~=~&- \frac{\partial F_3}{\partial p_i}, \cr P_i~=~&- \frac{\partial F_3}{\partial Q^i}, \cr K~=~&\lambda H+ \frac{\partial F_3}{\partial t} .\end{align} \tag{9.20'}$$

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