The exact SL(K+3,C) symmetry of string theory

By using on-shell recursion relation of string scattering amplitudes (SSA), we show that all n-point SSA of the open bosonic string theory can be expressed in terms of the Lauricella functions. This result extends the previous exact SL(K+3,C) symmetry of the 4-point Lauricella SSA (LSSA) of three tachyons and one arbitrary string states to the whole tree-level open bosonic string theory. Moreover, we present three applications of the SL(K+3,C) symmetry on the SSA. They are the solvability of all n-point SSA in terms of four-tachyon amplitudes, the existence of iteration relations among residues of a given SSA so as to soften its hard scattering behavior and finally the re-derivation of infinite linear relations among hard SSA [12].