Free boundary q-Whittaker and Hall-Littlewood processes

We study the free boundary $q$-Whittaker and Hall--Littlewood processes, two probability measures on sequences of partitions. We prove that a certain observable of the free boundary $q$-Whittaker process exhibits a $(q,t)$ symmetry after a random shift, generalizing a previous result of Imamura, Mucciconi, and Sasamoto, and an extension of that result due to the first author. Our proof is completely different, and as part of our proof, we find contour integral formulas for the free boundary $q$-Whittaker process. We also show a matching between certain observables in the free boundary Hall--Littlewood process and a quasi-open six vertex model, and explain how work of Finn and Vanicat gives an evaluation of a bounded sum over skew Hall--Littlewood functions as a rectangular Koornwinder polynomial.