The one-variable q-coherent states attached to the q-disc algebra are constructed and used to obtain the q-Bargmann–Fock realization of its Fock representation. Then, this realization is used to obtain the q−1-continuous Hermite polynomials as well as continuous and discrete q-Hermite polynomials by using a pair of Hermitian canonical conjugate operators and two pairs of the non-Hermitian conjugate operators, respectively. Besides, we introduce a two-variable family of q-coherent states attached to the Fock representation space of the q-disc algebra and its opposite algebra and obtain their simultaneous q-Bargmann–Fock realization. For an appropriate non-Hermitian operator, the latter realization is served to obtain the well-known little q-Jacobi polynomials used in constructing the q-disc polynomials.

中文翻译：

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附加到量子圆盘代数及其相关多项式的相干态

一变量q- 相干状态附加到q- 盘代数被构造并用于获得q-Bargmann–Fock 实现其 Fock 表示。然后，这个实现被用来获得q-1- 连续 Hermite 多项式以及连续和离散q-Hermite 多项式分别使用一对 Hermitian 规范共轭算子和两对非 Hermitian 共轭算子。此外，我们引入了一个二变量族q- 连接到 Fock 表示空间的相干状态q-圆盘代数及其相反代数并获得它们的同时q-Bargmann-Fock 实现。对于适当的非厄米算子，后一种实现用于获得众所周知的小q-Jacobi 多项式用于构造q-圆盘多项式。