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Symmetric discrete AKP and BKP equations

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