Empty Sum

In mathematics, an empty sum, or nullary sum, is a summation involving no terms at all. The value of any empty sum of numbers is conventionally taken to be zero. For summations defined in terms of addition of other values than numbers (such as vectors, matrices, polynomials), in general of values in some given Abelian group, the value of an empty summation is taken to be the zero element of that group.

An empty sum can arise in particular for expressions of the form

when ; in such case the summation has value 0 (or the zero element of the additive group in which the terms t_i live).

A situation similar to an empty summation can arise for other operations than addition; notably, its counterpart for multiplication is an empty product, a product of no factors at all. In such cases the convention of a zero result does not apply; indeed the value of an empty product of numbers is taken to be one, the neutral element for multiplication.