The $SL(K+3,\mathbb{C})$ symmetry of string scatterings from D-branes

By using the solvability of Lauricella function $F_{D}^{(K)}\left( \alpha;\beta_{1},…,\beta_{K};\gamma;x_{1},…,x_{K}\right) $ with nonpositive integer ${\beta }_J$, we show that each scattering or decay process of string and D-brane states at $\mathit{\text{arbitrary}}$ mass levels can be expressed in terms of a single Lauricella function. This result extends the previous exact $SL(K+3,\mathbb{C})$ symmetry of tree-level open bosonic string theory to include the D-brane.