  We develop a theory of enlarged mixed Shimura varieties, putting the
universal vectorial bi-extension defined by Coleman into this framework to
study some functional transcendental results of Ax type. We study their
bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its
relation with the Ax logarithmique theorem and the Ax-Lindemann theorem which
we shall prove. We also prove the whole Ax-Schanuel conjecture for the
unipotent part. All these bi-algebraic and transcendental results generalize
their counterparts for mixed Shimura varieties. In the end we briefly discuss
about the Andre-Oort and Zilber-Pink type problems for enlarged mixed Shimura
varieties.
