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  "body": "\n## Definition\n\nLet $a, b, c, d$ be real numbers with $c \\ne 0$ and $ad - bc \\ne 0$.\nThe *Riccati difference equation* with these coefficients is the [[rational difference equation|T8YDEASD1H]]\n$$x_{n+1} = \\frac{a x_n + b}{c x_n + d} \\qquad (n \\ge 0),$$\nfor real sequences $(x_n)_{n \\ge 0}$ with $c x_n + d \\ne 0$ for all $n$.\n\n## Remarks\n\nExample: $x_{n+1} = 1/(1 + x_n)$ has $a = 0$, $b = c = d = 1$, so $c \\ne 0$ and $ad - bc = -1 \\ne 0$.\nNon-example: $x_{n+1} = (2 x_n + 1)/3$ has $c = 0$; it is an affine linear recurrence, not a Riccati equation.\nThe condition $ad - bc \\ne 0$ excludes the degenerate case where the right side is a constant.\nLinearization: if $(y_n)$ satisfies $y_{n+2} = (a + d)\\, y_{n+1} - (ad - bc)\\, y_n$ with all $y_n \\ne 0$, then $x_n = (y_{n+1}/y_n - d)/c$ solves the equation.\nTo check, put $w_n = c x_n + d$; the equation becomes $w_{n+1} = (a + d) - (ad - bc)/w_n$, and $w_n = y_{n+1}/y_n$ turns this into the linear recurrence.\n",
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