In mathematics, a summation equation or discrete integral equation is an equation in which an unknown function appears under a summation sign. The theories of summation equations and integral equations can be unified as integral equations on time scales using time scale calculus. A summation equation compares to a difference equation as an integral equation compares to a differential equation. The Volterra summation equation is:
x ( t ) = f ( t ) + ∑ s = m n k ( t , s , x ( s ) ) {\displaystyle x(t)=f(t)+\sum _{s=m}^{n}k{\bigl (}t,s,x(s){\bigr )}}
where x is the unknown function, s, t are integers, and f, k are known functions.
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Summation equations or discrete integral equations