Proposition: Numerator representation

By Theorem thm:wpA-Sbv, Input (S-bv) holds. Then, using Lemma lem:bv(a),(b), ${\partial }^{\alpha }S=c{p}_{\alpha }{T}_{1+|\alpha |}$ on $\Omega$. Here $p_0=1$ and $p_{\alpha +e_{\rho }}=Q{\partial }_{\rho }{p}_{\alpha }-(1+|\alpha |)p_{\alpha }{\partial }_{\rho }Q$, so $p_{\alpha }$ is homogeneous of degree $|\alpha |$. For $\lambda =\{\alpha ,\beta \}$ of order $N=|\alpha |+|\beta |$, Lemma lem:bv(c) gives

Proof. By induction, ${\partial }_{\rho }(p_{\alpha }{T}_k)=({\partial }_{\rho }{p}_{\alpha })T_k-k{p}_{\alpha }({\partial }_{\rho }Q)T_{k+1}$ by (b), and $T_k=Q{T}_{k+1}$ by (a); this gives the recursion. The product is the Hörmander product used for $q^{\circ }$ on $y\ne w$ (XVI-J: “the pointwise product of two copies of ${{\Delta }_F}^{\mathrm{FR}}$ exists”), and (c) applies. kernel_relations.py computes $p_{\alpha }$ from the same recursion, realised as ${\partial }^{\alpha }(1/Q)=p_{\alpha }/Q^{1+|\alpha |}$, for all $|\alpha |\le 4$.