Hi,
> but can you give me some reference to this?
I think I pointed you to some references in the last mail.
Titles of some more or less important literature on the topic:
Summation in Finite Terms
Theory of Summation in Finite Terms
Algebraic Extensions for Summation in Finite Terms
A Refined Difference Field Theory for Symbolic Summation
An Implementation of Karr's Summation Algorithm in Mathematica
Product Representations in ΠΣ-Fields
Simplifying Sums in ΠΣ-Extensions
Solving Parameterized Linear Difference Equations in ΠΣ-Fields
Symbolic Summation with Single-Nested Sum Extensions
Structural theorems for symbolic summation
Symbolic Summation in Difference Fields
The Summation Package Sigma: Underlying Principles and a Rhombus Tiling Application
Just search for these articles, you should be able to find most
of them online.