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Abstract. In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction(adjoint eigenfunction) and the spectral representation of its Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.

In this paper, we construct the additional symmetries for the Toda lattice (TL) hierarchies of B type and C type (the BTL and CTL hierarchies), and show their algebraic structures are w∞ respectively. And also we discuss the generating functions of the additional symmetries.