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  "body": "\n## Definition\n\nLet $k \\ge 0$ be an integer, and let $\\alpha, \\beta_0, \\dots, \\beta_k, A, B_0, \\dots, B_k$ be real numbers.\nA *rational difference equation* is a [[nonlinear difference equation|7C7FBZGERM]] of the form\n$$x_{n+1} = \\frac{\\alpha + \\sum_{i=0}^{k} \\beta_i\\, x_{n-i}}{A + \\sum_{i=0}^{k} B_i\\, x_{n-i}} \\qquad (n \\ge 0),$$\nfor real sequences $(x_n)_{n \\ge -k}$.\nThe initial values $x_{-k}, \\dots, x_0$ are taken such that the denominator is nonzero for every $n \\ge 0$.\n\n## Remarks\n\nExample: $x_{n+1} = 1/(1 + x_n)$ is a rational difference equation with $k = 0$, $\\alpha = 1$, $\\beta_0 = 0$, $A = 1$, $B_0 = 1$.\nFor $x_0 = 1$ all denominators are at least $1$, and $x_n = F_{n+1}/F_{n+2}$ with $F$ the [[Fibonacci numbers|0GA9K57A39]].\nNon-example: $x_{n+1} = 1/(1 + x_n^2)$ is not of this form, since its denominator is not affine in $x_n$.\nIf all $B_i = 0$ and $A \\ne 0$, the equation is linear; the interesting cases have some $B_i \\ne 0$.\nThe 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]].\n",
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