  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 $1 \le i \le
d$.
