Continuous q-Hermite polynomials

In mathematics, the continuous q-Hermite polynomials  are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition
The polynomials are given in terms of basic hypergeometric functions by
 * $$H_n(x|q)=e^{in\theta}{}_2\phi_0\left[\begin{matrix}

q^{-n},0\\ -\end{matrix}
 * q,q^n e^{-2i\theta}\right],\quad x=\cos\,\theta.$$

Recurrence and difference relations

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

with the initial conditions


 * $$ H_0 (x\mid q) =1, H_{-1} (x\mid q) = 0$$

From the above, one can easily calculate:



\begin{align} H_0 (x\mid q) & = 1 \\ H_1 (x\mid q) & = 2x \\ H_2 (x\mid q) & = 4x^2 - (1-q) \\ H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\ H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1-q-q^3+q^4) \end{align} $$

Generating function

 * $$ \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}

{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty} $$ where $$\textstyle x=\cos \theta$$.