Appears in collection : Combinatorics and Arithmetic for Physics

The theory of $q$-Whittaker functions for classical types is known to have a (quantum) cluster algebra realization. In this framework, a natural connection with the quantum dilogarithm is known. We show how this extends to the more general case of Macdonald theory in type A. We propose new Givental-like and Mellin-Barnes-like expressions for the Macdonald functions, and explore their properties. These involve heavy use of quantum dilogarithms. (ongoing collaboration with M. Bershtein, J.-E. Bourgine, R. Kedem, V. Pasquier and J. Shiraishi).