A binomial expansion formula for weighted geometric means of unipotent matrices

Following the Kubo-Ando theory of operator means we consider the weighted geometric mean (Formula presented.) of (Formula presented.) upper triangular matrices A and B whose main diagonals are all 1, named the upper unipotent matrices. We also present its binomial expansion (Formula presented.) Showing that the weighted geometric mean is a geodesic of symmetry in the symmetric space equipped with point reflection, known as the Loos symmetric space, we derive several binomial identities on the Lie group of upper unipotent (resp. the Lie algebra of nilpotent) matrices.