We consider the double affine Hecke algebra
$H=H(k_0,k_1,k^\vee_0,k^\vee_1;q)$ associated with the root system
$(C^\vee_1,C_1)$. We display three elements $x,y,z$ in $H$ that satisfy
essentially the $Z_3$-symmetric Askey-Wilson relations. We obtain the relations
as follows. We introduce an algebra $\hat H$ that is more general than $H$,
called the universal double affine Hecke algebra of type $(C_1^\vee,C_1)$. An
advantage of $\hat H$ over $H$ is that it is parameter free and has a larger
automorphism group.

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
$E$.

Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with
finite positive dimension.