  By using pseudo-differential operators containing two derivations, we extend
the Kadomtsev-Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For
the KP-mKP hierarchy, we obtain its B\"{a}cklund transformations, bilinear
equations of Baker-Akhiezer functions and Hirota equations of tau functions.
Moreover, we show that this hierarchy is equivalent to a subhierarchy of the
dispersive Whitham hierarchy associated to the Riemann sphere with its infinity
point and one movable point marked.
