  We investigate the emergence and consequences of odd-frequency spin-triplet
s-wave pairing in superconducting hybrid junctions at the edge of a
two-dimensional topological insulator without any magnetism. More specifically,
we consider several different normal-superconductor hybrid systems at the
topological insulator edge, where spin-singlet s-wave superconducting pairing
is proximity-induced from an external conventional superconductor. We perform
fully analytical calculations and show that odd-frequency mixed spin-triplet
s-wave pairing arises due to the unique spin-momentum locking in the
topological insulator edge state and the naturally non-constant pairing
potential profile in hybrid systems. Importantly, we establish a one-to-one
correspondence between the local density of states (LDOS) at low energies and
the odd-frequency spin-triplet pairing in NS, NSN and SNS junctions along the
topological insulator edge; at interfaces the enhancement in the LDOS can
directly be attributed to the contribution of odd-frequency pairing.
Furthermore, in SNS junctions we show that the emergence of the zero-energy
LDOS peak at the superconducting phase $\phi=\pi$ is associated purely with
odd-frequency pairing in the middle of the junction.
