The Recursion Operator Of Integrable Hierarchies

So far,lots of attention have been given to the study of KP hierarchy in the field of integrable systems.Its Lax equation,wave equation,? function,additional symmetry and recursion operator have already been successively given,and different forms of promotion have been further studied.Among them,q-deformation modified-KP hierarchy(q-mKP hierarchy)and discrete mKP hierarchy(d-mKP hierarchy)have attracted much attention.The Lax equation,dressing operator,wave equation,additional symmetry under these two hierarchy have been researched.Recursion operator is one of the concerns of KP hierarchy,the recursion operator of KP hierarchy and q-KP hierarchy have been published.In this Letter,we would like to study the recursion operator of q-mKP hierarchy and d-mKP hierarchy.