4. Linear difference systems with singularity

4.1 Introduction and notations

Let's start with Example 7. The study of the function Γ is not limited to its integer values. We can consider extending this function to real or complex values of the argument, i.e. consider the function Γ defined on $\mathrm{C}$ and verifying the functional equation: Γ (x + 1) = xΓ (x ), whose asymptotic behavior is well studied.

We adopt this point of view in what follows, i.e. we now study difference equations whose unknown is a function of a complex variable. The translation operator is