Empty sum

This satisfies the recurrence provided that we use the following natural convention:

Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas.

Such "sums" are natural starting points in induction proofs, as well as in algorithms.

For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a zero element).

For the same reason, the empty product is taken to be the multiplicative identity.

For sums of other objects (such as vectors, matrices, polynomials), the value of an empty summation is taken to be its additive identity.

The empty sum convention allows the zero-dimensional vector space V={0} to have a basis, namely the empty set.