  We extend the notion of a spectral triple to that of a higher-order relative
spectral triple, which accommodates several types of hypoelliptic differential
operators on manifolds with boundary. The bounded transform of a higher-order
relative spectral triple gives rise to a relative K-homology cycle. In the case
of an elliptic differential operator on a compact smooth manifold with
boundary, we calculate the K-homology boundary map of the constructed relative
K-homology cycle to obtain a generalization of the Baum-Douglas-Taylor index
theorem.
