标题： 双边$q$-超球函数 显示英文标题

标题： Bilateral $q$-ultraspherical functions

摘要： 我们引入了连续$q$-超球多项式的双边扩展，我们称之为双边$q$-超球函数。 这些函数作为特定的双边基本超几何${}_2\psi_2$系列给出，它们在变量$x=\cos\theta$中是解析的，并且依赖于两个参数$\beta$和$\gamma$以及一个基$q$。 对于这些双边$q$-超球函数，我们推导出一个双边生成函数，找到一个三项递推关系，解释它们在 Askey--Wilson 分差算子作用下的行为，并证明它们满足一种移位正交性。 显示英文摘要

摘要： We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2\psi_2$ series, they are analytic in a variable $x=\cos\theta$ and depend on two parameters $\beta$ and $\gamma$ and on a base $q$. For these bilateral $q$-ultraspherical functions we derive a bilateral generating function, find a three-term recurrence relation, explain how they behave under the action of the Askey--Wilson divided difference operator, and show that they satisfy a type of shifted orthogonality.