  We generalize the Toda lattice hierarchy by considering N+M dependent
variables. We construct roots and logarithms of the Lax operator which are
uniquely defined operators with coefficients that are $\epsilon$-series of
differential polynomials in the dependent variables, and we use them to provide
a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix
theory we give the bihamiltonian formulation of this hierarchy and we prove the
existence of a tau function for its solutions. Finally we study the
dispersionless limit and its connection with a class of Frobenius manifolds on
the orbit space of the extended affine Weyl groups of the $A$ series.
