标题： 终止表示，变换和求和对于Askey--Wilson多项式的$q$和$q^{-1}$对称子族 显示英文标题

标题： Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials

摘要： 在本文中，我们详尽地探讨了Askey--Wilson多项式的$q$和$q^{-1}$-对称子族的终止基本超几何表示和变换。 这些子族是通过反复将一个自由参数（不是$q$）设为零，直到没有参数剩余而得到的。 这些子族（以及它们的$q^{-1}$对应项）是连续对偶$q$-Hahn多项式、Al-Salam--Chihara多项式、连续大$q$-Hermite多项式和连续$q$-Hermite多项式。 从这些多项式的终止基本超几何表示出发，并由于其自由参数的对称性，我们能够详尽地探讨这些多项式所满足的终止基本超几何变换公式。 然后，我们研究由这些多项式的终止表示所暗示的终止变换结构。 我们最后描述$q$-Askey 模式对称群结构。 显示英文摘要

摘要： In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly setting one of the free parameters (not $q$) equal to zero until no parameters are left. These subfamilies (and their $q^{-1}$ counterparts) are the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite, and the continuous $q$-Hermite polynomials. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulas which these polynomials satisfy. We then study the terminating transformation structure which are implied by the terminating representations of these polynomials. We conclude by describing the symmetry group structure of the $q$-Askey scheme.