$$ \begin{align} v_\pi(s) := & \mathbb{E}\pi [G_t|S_t=s] \ = & \mathbb{E}\pi [R_{t+1}+\gamma G_{t+1}|S_t=s] \ = & \sum_a \pi(a|s) \sum_{s'}\sum_r p(s',r|s,a)[r+\gamma\mathbb{E}\pi [G{t+1}|S_{t+1}=s']] \ = & \sum_a \pi(a|s) \sum_{s',r} p(s',r|s,a)[r+\gamma v_\pi(s')] \end{align} $$
$$ \begin{align} q_\pi(s,a) := & \mathbb{E}\pi [G_t|S_t=s,A_t=a] \ = & \mathbb{E}\pi [R_{t+1}+\gamma G_{t+1}|S_t=s,A_t=a] \ = & r(s,a) + \gamma \sum_{s',a'} \pi(a'|s') p(s'|s,a) q_\pi(s',a') \ = & \sum_{s',r}rp(s',r|s,a) + \gamma \sum_{s',a',r} p(s',r|s,a) \pi(a'|s') q_\pi(s',a') \end{align} $$
$$ v_\pi(s) = \mathbb{E}\pi q\pi(s, A) = \sum_a \pi(a|s) q_\pi(s,a) $$
$$ q_(s,a) = \mathbb{E} [R_{t+1} + \gamma v_(S_{t+1})|S_t=s, A_t=a] $$
$$ \begin{align} v_(s) =& \max_{a\in\mathcal{A}(s)} q_{\pi_}(s,a) \ =& \max_a \mathbb{E}{\pi} [G_t | S_t=s, A_t=a] \ =& \max_a \mathbb{E}{\pi} [R_{t+1} + \gamma G_{t+1} | S_t=s, A_t=a] \ =& \max_a \mathbb{E} [R_{t+1} + \gamma v_(S_{t+1}) | S_t=s, A_t=a] \ =& \max_a \sum_{s',r} p(s',r|s,a) [r+\gamma v_(s')] \end{align} $$
$$ \begin{align} q_(s,a) =& \mathbb{E} [R_{t+1}+\gamma \max_{a'} q_(S_{t+1},a')|S_t=s, A_t=a] \ =& \sum_{s',r} p(s',r|s,a)[r+\gamma \max_{a'} q_*(s',a')] \end{align} $$