The determinant representation of the gauge transformation for the discrete KP hierarchy

A successive gauge transformation operator $T_{n+k}$ for the discrete KP(dKP) hierarchy is defined, which is involved with two types of gauge transformations operators. The determinant representation of the $T_{n+k}$ is established,and then it is used to get a new $tau$ function $τ^{(n+k)}_\triangle$ of the dKP hierarchy from an initial $τ_\triangle$. In this process, we introduce a generalized discrete Wronskian determinant and some useful properties of discrete difference operator.