We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale T, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for T=R and T=Z within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) M(lambda) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

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