This paper regroups several results in the continuation of my D.E.A. report. This paper was originally intended to be part of my PhD. thesis. However, because my thesis has grown in size more than expected, I decided to suppress this part, which is independent from the rest of my thesis.

The first chapter concerns the computation with special functions determined by algebraic differential equations and intial conditions. The second part deals with a generalization of effective differential elimination theory to the context of so called D-rings, introduced by Nichols and Weisfeiler. The last, and most original part deals with a generalization of the results of chapter 2 to more general mixed differential-difference equations.

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Note: it was pointed out by François Boulier that the generalization of Rosenfeld's lemma does not work in the case of difference equations. A new approach has been developed Xiao Shan Gao and his students, who have given effective zero-tests for non-linear difference algebra.