The Lauricella functions and exact string scattering amplitudes

We discover that the 26D open bosonic string scattering amplitudes (SSA) of three tachyons and one arbitrary string state can be expressed in terms of the D-type Lauricella functions with associated S L (K+ 3 , ℂ) symmetry. As a result, SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These include the linear relations first conjectured by Gross [1-5] and later corrected and proved in [6-12] in the hard scattering limit, the recurrence relations in the Regge scattering limit with associated S L (5 ℂ) symmetry [24-26] and the extended recurrence relations in the nonrelativistic scattering limit with associated S L (4 ℂ) symmetry [29] discovered recently. Finally, as an application, we calculate a new recurrence relation of SSA which is valid for all energies.