Hirota Quadratic Equations For The Special Extended Bigraded Toda Hierarchy

The extended bigraded toda hierarchy(shortly EBTH)was introduced by Gudio Carlet[C]who hoped that the EBTH might be relevent for some applications in 2D topoligical field theory and in the theory of Gromov-Witten invariants.In[C],the lax-type presentation of the EBTH was given.In the paper of Todor.E.Milanov and Hsian-Hua Tseng[TH],they described conjecturally HQE(Hirota Quadratic Equation) of the EBTH and proved that it governs the Gromov-Witten theory of c_km.But the HQE are known only conjecturally.In this paper we describe the HQE of EBTH when N=M i.e.the Special Extended Bigraded Toda Hierarchy(shortly SEBTH)in terms of vertex operators.