The Q-polynomial idempotents of a distance-regular graph

We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d⩾3. Let E denote a minimal idempotent of Γ which is not the trivial idempotent E0. Let denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E∘E is a linear combination of E0, E, and at most one other minimal idempotent of Γ; (ii) there exists a complex scalar β such that is independent of i for 1⩽i⩽d−1; (iii) for 1⩽i⩽d.