Continuous big q-Hermite polynomials

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Short description: A family of basic hypergeometric orthogonal polynomials

In mathematics, the continuous big 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. [math]\displaystyle{ H_n(x;a|q)=a^{-n}{}_{3}\phi_2\left[\begin{matrix} q^{-n},ae^{i\theta},ae^{-i\theta}\\ 0,0\end{matrix} ;q,q\right],\quad x=\cos\,\theta. }[/math]

References