mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

appears under a

summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...

sign. The theories of summation equations and

integral equation In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...

s can be unified as ''integral equations on time scales''Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains
Tomasia Kulik, Christopher C. Tisdell, September 3, 2007 using

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 hy ...

. A summation equation compares to a

difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

as an

compares to a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

. The Volterra summation equation is: :

x(t) = f(t) + \sum_^n k(t, s, x(s))

where ''x'' is the unknown function, and s, a, t are integers, and f, k are known functions.

References

Summation equations or discrete integral equations
Integral equations {{mathanalysis-stub