1}^p k_h$. We just need to sum up over the index $h$ in $(3)$ and we can read out the expressions for the intercept, main effects and interactions up to the $Q^{th}$ order. In particular, we have that the intercept is equal to $\sum_{h = 1}^k \sum_{q = 1}^{\lceil \frac{Q-1}{2} \rceil } \omega_{h}^{(2q)} V^{2 q }_{hh} b_{q0}^e $. When $Q = 2$ this reduces to $\sum_{h = 1}^k \omega_{h}^{(2)} V^{2  }_{hh}  = tr(\Omega V) $, where $\Omega = diag(\omega_1^{(2)},\cdots,\omega_k^{(2)})$. The expression for the main effects coefficients on $X_j$ is $ \sum_{h = 1}^k \sum_{q = 1}^{\lfloor \frac{Q+1}{2} \rfloor } \omega_{h}^{(2q-1)} V^{2 q -1 }_{hh} b_{q1}^o a_{hj} $. When $Q = 2$ this becomes $ \sum_{h= 1}^k \omega_{h}^{(1)} a_{hj}$, hence $\beta_X = A^T \omega$. Similarly the expression for the interaction between $X_j$ and $X_l$ is equal to $ \sum_{h = 1}^k \sum_{q = 1}^{\lceil \frac{Q-1}{2} \rceil  } 2 \omega_{h}^{(2q)} V^{2 q }_{hh} b_{q1}^e a_{hj}  a_{hl}$ and when $Q = 2$ we have $\sum_{h = 1}^k 2 \omega_{h}^{(2)} a_{hj}  a_{hl}$ which is equal to $2 [A^T \Omega A]_{(j,l)}$.

In general, if we are interested in the $q^{th}$ order interactions, we can find the expression on the top summation for $f = \frac{q+1}{2}$ 