In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity.

Let , , , ... be a sequence of numbers, and let
:be the sum of the first m terms of the sequence. This satisfies the recurrence
:provided that we use the following natural convention: .In other words, a "sum" with