  We explore the idea of higher-order flavor sum rules, i.e. sum rules that
hold to higher orders in the flavor symmetry-breaking parameters. In
particular, we consider isospin sum rules based on $B \to n\pi$ decays, and
U-spin sum rules based on $B_d$ and $B_s$ decays to $n$ charged pions and/or
kaons. We also demonstrate that sum rules exist to arbitrary orders by
considering decay modes with increasing number of final state mesons $n$.
Finally, we identify various theoretical and practical issues that need to be
addressed in utilizing these sum rules.
