Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy

We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlevé equation $\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations admit the limit to the first Painlevé equation. It is shown that each of these equations describes Bäcklund transformations of Veselov–Shabat periodic dressing lattices with odd period known also as Noumi–Yamada systems of type $A_{2(n-1)}^{(1)}$.