Equivalent T-Q relations and exact results for the open TASEP

Starting from the Bethe ansatz solution for the open Totally Asymmetric Simple Exclusion Process (TASEP), we compute the largest eigenvalue of the deformed Markovian matrix, in exact agreement with results obtained by the matrix ansatz. We also compute the eigenvalues of the higher conserved charges. The key step is to find a simpler equivalent T-Q relation, which is similar to the one for the TASEP with periodic boundary conditions.