Continuous q-Hermite polynomials
# Continuous q-Hermite polynomials

In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) 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

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

with the initial conditions

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

From the above, one can easily calculate:

    ${\begin{aligned}H_{0}(x\mid q)&=1\\H_{1}(x\mid q)&=2x\\H_{2}(x\mid q)&=4x^{2}-(1-q)\\H_{3}(x\mid q)&=8x^{3}-2x(2-q-q^{2})\\H_{4}(x\mid q)&=16x^{4}-4x^{2}(3-q-q^{2}-q^{3})+(1-q-q^{3}+q^{4})\end{aligned}}$

## Generating function

    $\sum _{n=0}^{\infty }H_{n}(x\mid q){\frac {t^{n}}{(q;q)_{n}}}={\frac {1}{\left(te^{i\theta },te^{-i\theta };q\right)_{\infty }}}$

where $\textstyle x=\cos \theta$.

