The simplification can be done by called Mathematica's simplifier:
Now we show another example, wich shows the mathematical power of the Gosper-Zeilberger-Paule-prover: The problem of constructing a closed form for the input expression in this example was an open problem for many years. The solution was given in P. Paule, Computer-Solution of Problem 94-2, SIAM REVIEW Vol.37 (1995), 105-106.
Again, the simplification can be accomplished by calling Mathematica's simplifier: