Symmetric discrete AKP and BKP equations

Abstract We show that when KP (Kadomtsev–Petviashvili) τ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of AKP and its reflected symmetric forms. Thus equations share same with these symmetries. Such connection is extended to 4 dimensional (i.e. higher order) in corresponding hierarchies. Various explicit forms such functions, including Hirota’s form, Gramian, Casoratian polynomial, are given. Symmetric Cauchy matrix form composed Weierstrass σ investigated. As result we obtain elliptic coefficients.