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We introduce Baxter Q-operators for the quantum Ruijsenaars hyperbolicsystem. We prove that they represent a commuting family of integral operatorsand also commute with Macdonald difference operators, which are gaugeequivalent to the Ruijsenaars Hamiltonians of the quantum system. The proof ofcommutativity of the Baxter operators uses a hypergeometric identity onrational functions that generalize Ruijsenaars kernel identities.

Baxter operators in Ruijsenaars hyperbolic system I. Commutativity of Q-operators