论文标题
$ q $ - Q $ Q $ -Laguerre polyenmials和Little $ Q $ -Jacobi多项式的注释
Notes on $q$-partial differential equations for $q$-Laguerre polynomials and little $q$-Jacobi polynomials
论文作者
论文摘要
我们定义两个常见的$ q $ - 正交多项式：均质$ q $ -laguerre多项式和均质的小$ q $ -Jacobi多项式。可以将它们分别视为两个$ Q $ - 优势差异方程式的解决方案。然后，我们证明，如果一个分析函数满足$ Q $ - 优势差异方程的某个系统，并且仅当它可以根据均质$ q $ -laguerre polyentorials或均质的小$ q $ $ q $ -JACOBI多项式而进行扩展。作为应用程序，我们获得了Ramanujan $ Q $ -Beta积分和Andrews-Askey积分的概括。此外，我们提出了$ q $ -laguerre多项式的运营商表示，该代表促进了涉及$ q $ laguerre多项式的身份的计算。
We define two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Then, we proved that if an analytic function satisfies a certain system of $q$-partial differential equations, if and only if it can be expanded in terms of homogeneous $q$-Laguerre polynomials or homogeneous little $q$-Jacobi polynomials. As applications, we obtain generalizations of the Ramanujan $q$-beta integrals and Andrews-Askey integrals. Additionally, we present an operator representation of $q$-Laguerre polynomials that facilitates the computation of identities involving $q$-Laguerre polynomials.
