# recurrence

**recurrence** A statement describing some quantity such as *f*(*n*) (where *f* is some function and *n* is a positive integer) in terms of values of *f*(*m*), where *m* is a nonnegative integer smaller than *n*; initial values such as *f*(0) or *f*(1) can be assumed to be defined. The concept can be extended to include functions of several variables. A recurrence will then involve defining *f*(*m*,*n*), say, in terms of *f*(*m*′,*n*′) where in some sense (*m*′,*n*′) is smaller than (*m*,*n*); again initial values can be assumed. The numbers in the Fibonacci series can be defined by a recurrence.

In general, a recurrence can be considered as an equation connecting the values of the function at a number of related points. It has the form *g*(*n*, *f*(*n*), *f*(*n*–1),…, *f*(*n*–*k*)) = 0 *n* = *k*, *k* + 1,…, *N*

Assuming initial values for *f*(0), *f*(1),…, *f*(*k*–1), values for other points *n* can be calculated.

Equations of this type arise naturally in the discretization of continuous problems, and in a slightly different form, known as a difference equation, appear repeatedly in combinatorics.

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# Recurrence (Disease)

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**Recurrence (Disease)**