---
tag: 49837NYG1A
kind: definition
name: Rational difference equation
msc: [39A20, 39A05]
defines: T8YDEASD1H
uses: [7C7FBZGERM]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_difference_equation
---

## Definition

Let $k \ge 0$ be an integer, and let $\alpha, \beta_0, \dots, \beta_k, A, B_0, \dots, B_k$ be real numbers.
A *rational difference equation* is a [[nonlinear difference equation|7C7FBZGERM]] of the form
$$x_{n+1} = \frac{\alpha + \sum_{i=0}^{k} \beta_i\, x_{n-i}}{A + \sum_{i=0}^{k} B_i\, x_{n-i}} \qquad (n \ge 0),$$
for real sequences $(x_n)_{n \ge -k}$.
The initial values $x_{-k}, \dots, x_0$ are taken such that the denominator is nonzero for every $n \ge 0$.

## Remarks

Example: $x_{n+1} = 1/(1 + x_n)$ is a rational difference equation with $k = 0$, $\alpha = 1$, $\beta_0 = 0$, $A = 1$, $B_0 = 1$.
For $x_0 = 1$ all denominators are at least $1$, and $x_n = F_{n+1}/F_{n+2}$ with $F$ the [[Fibonacci numbers|0GA9K57A39]].
Non-example: $x_{n+1} = 1/(1 + x_n^2)$ is not of this form, since its denominator is not affine in $x_n$.
If all $B_i = 0$ and $A \ne 0$, the equation is linear; the interesting cases have some $B_i \ne 0$.
The case $k = 0$, $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \ne 0$ and $ad - bc \ne 0$, is the [[Riccati difference equation|D4Q8VM7826]].