VALUES OF E-FUNCTIONS ARE NOT LIOUVILLE NUMBERS

Shidlovskii has given a linear independence measure of values of E -functions with rational Taylor coefficients at a rational point, not a singularity of the underlying differential system satisfied by these E -functions. Recently, Beukers has proved a qualitative linear inde-pendence theorem for the values at an algebraic point of E -functions with arbitrary algebraic Taylor coefficients. In this paper, we obtain an analogue of Shidlovskii's measure for values of arbitrary E -functions at algebraic points. This enables us to solve a long standing problem by proving that the value of an E -function at an algebraic point is never a Liouville number. We also prove that values at rational points of E -functions with rational Taylor coefficients are linearly independent over Q if and only if they are linearly independent over Q. Our methods rest upon improvements of results obtained by Andre and Beukers in the theory of E -operators.