The continuous $q-$Hermite polynomials are defined by

$${H_{n + 1}}(x|q) = 2x{H_n}(x|q) +( {q^n}-1){H_{n - 1}}(x|q)$$

with initial values ${H_{ - 1}}(x|q) = 0$ and ${H_0}(x|q) = 1.$

Cf. e.g. http://aw.twi.tudelft.nl/~koekoek/askey/ch3/par26/par26.html

The only simple special values I know of are $$\sqrt {{q^n}} {H_n}\left( {\frac{1}{2}\left( {\sqrt q + \frac{1}{{\sqrt q }}} \right),1,{q^2}} \right) = (1 + q)(1 + {q^2}) \cdots (1 + {q^n}).$$ I have seen this formula in the literature but do not remember where. Could you please give me some references? Are there other simple special values known?