A characterization of Q-polynomial distance-regular graphs

We obtain the following characterization of $Q$-polynomial distance-regular graphs. Let $\G$ denote a distance-regular graph with diameter $d\ge 3$. Let $E$ denote a minimal idempotent of $\G$ which is not the trivial idempotent $E_0$. Let $\{\theta_i^*\}_{i=0}^d$ denote the dual eigenvalue sequence for $E$. We show that $E$ is $Q$-polynomial if and only if (i) the entry-wise product $E \circ E$ is a linear combination of $E_0$, $E$, and at most one other minimal idempotent of $\G$; (ii) there exists a complex scalar $\beta$ such that $\theta^*_{i-1}-\beta \theta^*_i + \theta^*_{i+1}$ is independent of $i$ for $1 \le i \le d-1$; (iii) $\theta^*_i \ne \theta^*_0$ for $1 \le i \le d$.