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

Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction