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.

The polynomials are given in terms of basic hypergeometric functions. ${\displaystyle H_{n}(x;a|q)=a^{-n}{}_{3}\phi _{2}\left[{\begin{matrix}q^{-n},ae^{i\theta },ae^{-i\theta }\\0,0\end{matrix}};q,q\right],\quad x=\cos \,\theta .}$

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• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN  978-0-521-83357-8, MR  2128719
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