Continuous q-Hermite polynomials

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.

References