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As far as I know, the best relation between the two is the following: the field generated by the hilbert class polynomial $h_\mathcal{O} (X)$ contains the field generated by $h_K(X)$. This is implied by Proposition 25 of http://www.math.uga.edu/~pete/torspaper.pdf This implies among other things that $\deg(h_K(X)) | \deg(h_\mathcal{O}(X))$ (although this could be determined by simpler means).

Now as to your question about whether one can be generated from the other? No, unless you're in a very limited set of circumstances like $\deg(h_\mathcal{O}(X)) =1$ or such a thing. In fact it's a celebrated theorem of Heilbronn that $\deg(h_\mathcal{O}(X)) \to \infty$ as $|D| \to \infty$.infty$ where $D$ is the discriminant of $\mathcal{O}$.
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As far as I know, the best relation between the two is the following: the field generated by the hilbert class polynomial $H(D)$ h_\mathcal{O} (X)$ contains the field generated by $H(D_0)$ where $D_0$ is the discriminant of $\mathcal{O}_K$ and $D$ is the discriminant of $\mathcal{O}$. h_K(X)$. This is implied by Proposition 25 of http://www.math.uga.edu/~pete/torspaper.pdf This implies among other things that $h(D_0) \deg(h_K(X)) | h(D)$ \deg(h_\mathcal{O}(X))$ (although this could be determined by simpler means).

Now as to your question about whether one can be generated from the other? No, unless you're in a very limited set of circumstances like $h(D) \deg(h_\mathcal{O}(X)) =1$ or such a thing. In fact it's a celebrated theorem of Heilbronn that $h(D) \deg(h_\mathcal{O}(X)) \to \infty$ as $|D| \to \infty$.
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As far as I know, the best relation between the two is the following: the field generated by the hilbert class polynomial $H(D)$ contains the field generated by $H(D_0)$ where $D_0$ is the discriminant of $\mathcal{O}_K$ and $D$ is the discriminant of $\mathcal{O}$. This is implied by Proposition 25 of http://www.math.uga.edu/~pete/torspaper.pdf This implies among other things that $h(D_0) | h(D)$ (although this could be determined by simpler means).

Now as to your question about whether one can be generated from the other? No, unless you're in a very limited set of circumstances like $h(D) =1$ or such a thing. In fact it's a celebrated theorem of Heilbronn that $h(D) \to \infty$ as $|D| \to \infty$.