# Special values of continuous q - Hermite polynomials

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 ${H_0}(x|q) = 1.$

The only simple special values I know of are $$\sqrt {{q^n}} {H_n}\left( {\frac{1}{2}\left( {\sqrt q + \frac{1}{{\sqrt q }}} \right)|{q^2}} \right) = (1 + q)(1 + {q^2}) \cdots (1 + {q^n}).$$ 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?

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Why are there 3 variables inside $H_n$ in your special-value formula ? –  F. C. Jun 7 at 15:27
Sorry, I have corrected it. –  Johann Cigler Jun 7 at 19:50