On the extended 2-dimensional Toda lattice models

An extended $2$-dimensional Toda lattice equation is investigated by means of the Cauchy matrix approach. We introduce a direction parameter in the extension and represented the equation as a coupled system in a $3$-dimensional space. The equation can also be considered as a negative-order member in one direction of the discrete Kadomtsev–Petviashvili equation. By introducing the $\tau$-function and an auxiliary direction, the equation can be bilinearized in a $4$-dimensional space with a single $\tau$-function.