Constraints of the D$\mathrm{\Delta }$KP hierarchy to the semi-discrete AKNS and Burgers hierarchies

The paper investigates three eigenfunction constraints of two (2+1)-dimensional differential-difference integrable systems. First, we revisit the known squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$\mathrm{\Delta }$KP) hierarchy, which gives rise to a semi-discrete Ablowitz-Kaup-Newell-Segur hierarchy. Second, we introduce a linear eigenfunction constraint for the D$\mathrm{\Delta }$KP system and obtain a combined semi-discrete Burgers (sdBurgers) hierarchy. In the third one, we consider another linear eigenfunction constraint for the modified D$\mathrm{\Delta }$KP system and obtain the same combined sdBurgers hierarchy. All these constraint results are proved by using recursive algebraic structures of the involved integrable hierarchies generated by their master symmetries.