An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of $A$-type, and the Euler transformation formula for Kajihara and Noumi's multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto-Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the $SL(2,\mathbb{Z})$ duality of the DIM algebra.

中文翻译：

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福克张量空间上的广义麦克唐纳函数和改变首选方向的对偶公式

对于$N$-fold Fock 张量空间上的广义Macdonald 函数，获得了显式公式，计算多个筛选顶点算子的组合的某个矩阵元素。作为一个应用，我们证明了与 Ding--Iohara--Miki 代数（DIM 代数）相关的多价交织算子（或细化拓扑顶点算子）的任意矩阵元素的分解性质关于广义 Macdonald 函数，这是由 Awata、Feigin、Hoshino、Kanai、Yanagida 和作者之一推测的。我们的证明基于 $A$ 型普通 Macdonald 算子的渐近本征函数的组合和解析性质，以及 Kajihara 和 Noumi 的多重基本超几何级数的欧拉变换公式。