Algebraic properties of q-difference operators and q-summable functions

 One calls q-difference operator any expression of the form $a_0+a_1\sigma_q+...+a_n\sigma_q^n$, where q is some given constant different from zero and one, $a_0$, $...$, $a_n$ are known functions and where $\sigma_q$ denotes the q-schift operator defined by the relation $\sigma_q f(x)=f(qx)$. Such operators, appearing often in combinatorics or number theory, may be viewed as q-analog of ordinary differential operators. In our talk, we will see how to write everyone of such operators  as a product of a finite number of first order operators in some appropriate analytic setting. This factorization permits to define a q-summation procedure for solving the corresponding q-difference equation.