The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda systems Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Giventals integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmuller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-OConnell-Seppalainen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.
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