  The article addresses the problem whether indefinite double sums involving a
generic sequence can be simplified in terms of indefinite single sums.
Depending on the structure of the double sum, the proposed summation machinery
may provide such a simplification without exceptions. If it fails, it may
suggest a more advanced simplification introducing in addition a single nested
sum where the summand has to satisfy a particular constraint. More precisely,
an explicitly given parameterized telescoping equation must hold. Restricting
to the case that the arising unspecified sequences are specialized to the class
of indefinite nested sums defined over hypergeometric, multi-basic or mixed
hypergeometric products, it can be shown that this constraint is not only
sufficient but also necessary.
