Moments of the free Jacobi process: a matrix approach

We compute the large size limit of the moment formula derived in for the Hermitian Jacobi process at fixed time. In particular, we identify the terms contributing to the limit and show they satisfy a double recurrence relation. We also determine explicitly some of them and revisit a special case relying on Carlitz summation identity for terminating 1-balanced _4F_3 functions taken at unity.