The solution is not unique; it is determined only up to an additive [[periodic function]] with period 1. Therefore, each indefinite sum represents a family of functions.

The solution is not unique; it is determined only up to an additive [[periodic function]] with period 1. Therefore, each indefinite sum represents a family of functions.

The [[Niels Erik Nørlund|Nørlund]] principal solution represents the analytic solution without any such periodic terms. Two conventions exist, one for the forward difference, $\backslash Delta$, and one for the backward difference, $\backslash nabla$. The inverse forward difference, denoted $\backslash Delta^\{-1\}$, naturally extends the summation up to $x-1$. The inverse backward difference, denoted $\backslash nabla^\{-1\}$, naturally extends the summation up to $x$.

The [[Niels Erik Nørlund|Nørlund]] principal solution represents the analytic solution without any such non-constant periodic terms. Two conventions exist, one for the forward difference, $\backslash Delta$, and one for the backward difference, $\backslash nabla$. The inverse forward difference, denoted $\backslash Delta^\{-1\}$, naturally extends the summation up to $x-1$. The inverse backward difference, denoted $\backslash nabla^\{-1\}$, naturally extends the summation up to $x$.

==Fundamental theorem of the calculus of finite differences==

==Fundamental theorem of the calculus of finite differences==