In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a summation equation or discrete integral equation is an equation in which an unknown function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 appears under a summation

Summation

Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...

 sign. The theories of summation equations and integral equation

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

s can be unified as integral equations on time scales using time scale calculus

Time scale calculus

In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems...

. A summation equation compares to a difference equation as an integral equation

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

 compares to a differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

.

The Volterra summation equation is:
where x is the unknown function, and s, a, t are integers, and f, k are known functions.