Empty Sum - Relevance of Defining Empty Sums

Relevance of Defining Empty Sums

The necessity to define a value for empty summations is not immediately obvious, as it may appear strange to set up a summation when there is nothing to add. However empty summations often arise implicitly when the range of values being added depends on certain unknown parameters, and may become empty for certain values of the parameters. Leaving the value of an empty summation undefined would make it hard to make certain definitions properly, requiring frequent consideration of special cases to avoid empty sums. It would also imply an additional effort required in the proof of any statement involving summations to ensure that they never involve empty summations. In fact the reasons for defining empty sums are very similar to the reasons to considering things like the number zero and the empty set in the first place: while they seem to represent quite uninteresting notions, their existence allows for a much cleaner mathematical presentation of many subjects.

An even stronger case for the definition of empty sums arises when certain notions are defined in terms the existence of summations; not defining empty sums would implicitly alter such notions in a way that is usually undesirable. For instance in number theory, a partition of an integer n can be defined as a weakly decreasing sequence of positive integers whose sum is n. It is important that there is exactly one partition of the number 0 (most formulas for counting partitions would break down if there were assumed to be no partitions of 0), and since terms must be positive, the empty sequence is the only possible candidate. More generally, it often happens in combinatorics that certain values are considered to be a member of a given class by virtue of an empty sum, for instance 0 is a triangular number because of