  Let $\lambda_j$ be the $j$-th eigenvalue of Sturm-Liouville systems with
separated boundary conditions, we build up the Hill-type formula, which
represent $\prod\limits_{j}(1-\lambda_j^{-1})$ as a determinant of finite
matrix. This is the first attack on such a formula under non-periodic type
boundary conditions. Consequently, we get the Krein-type trace formula based on
the Hill-type formula, which express $\sum\limits_{j}{1\over \lambda_j^m}$ as
trace of finite matrices. The trace formula can be used to estimate the
conjugate point alone a geodesic in Riemannian manifold and to get some
infinite sum identities.
