# Continuous q-Hermite polynomial

The continuous $q$-Hermite polynomials are defined by $$\left\{ \begin{array}{ll} H_0(x|q)=1 \\ H_1(x|q)=2x \\ H_{n+1}(x|q) = 2xH_n(x|q) - (1-q^n)H_{n-1}(x|q). \end{array} \right.$$

# Properties

Theorem: The following formula holds: $$\dfrac{1}{(te^{i\theta};q)_{\infty}(te^{-i\theta};q)_{\infty}} = \displaystyle\sum_{k=0}^{\infty} H_k(\cos \theta|q) \dfrac{t^k}{(q;q)_k},$$ where $(\xi,q)_{\infty}$ denotes the $q$-Pochhammer symbol and $H_k(\xi|q)$ denotes a continuous $q$-Hermite polynomial.

Proof:

Theorem: The following formula holds: $$H_n(-x|q)=(-1)^nH_n(x|q).$$

Proof:

# References

On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials
Classical and quantum orthogonal polynomials in one variable by Ismail Mourad