Greatest Factorial Factorization and Symbolic Summation

The greatest factorial factorization (GFF) of a polynomial provides an analogue to square-free factorization but with respect to integer shifts instead to multiplicities. We illustrate the fundamental role of that concept in the context of symbolic summation. Besides a detailed discussion of the basic GFF notions we present a new approach to the indefinite rational summation problem as well as to Gosperʹs algorithm for summing hypergeometric sequences.