Summation equation

{{short description|Equation composed of a function being summed}}

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[http://web.maths.unsw.edu.au/~cct/tis-tomasia-IJDE-rev.pdf 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. A summation equation compares to a difference equation as an integral equation compares to a differential equation.

The Volterra summation equation is:

$$x(t)\; =\; f(t)\; +\; \backslash sum\_\{s=m\}^n\; k\; \backslash bigl(\; t,\; s,\; x(s)\; \backslash bigr)$$

where {{mvar|x}} is the unknown function, {{math|s, t}} are integers, and {{math|f, k}} are known functions.