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  "body": "\n## Definition\n\nLet $X$ be a set and let $(u_n)_{n \\ge 0}$ denote an unknown sequence in $X$.\nConsider a difference equation of the form\n$$F(n, u_n, u_{n+1}, \\dots, u_{n+k}) = 0 \\qquad (n \\ge 0),$$\nwhere $k \\ge 0$ is an integer and $F$ is a given function.\nThe equation has *order* $k$ if $F$ depends on both its second argument $u_n$ and its last argument $u_{n+k}$.\nFor a [[recurrence relation|07TZV87Y0T]] $u_{n+k} = f(n, u_n, \\dots, u_{n+k-1})$ in which $f$ depends on $u_n$, the order is $k$.\n\n## Remarks\n\nExample: $u_{n+3} = u_{n+1} + 2u_n$ has order $3$.\nThe indices that occur are $n$, $n+1$ and $n+3$, and the largest minus the smallest is $3$.\nNon-example: $u_{n+3} = 2u_{n+2}$ looks like order $3$, but $u_n$ does not occur.\nAfter the substitution $v_n = u_{n+2}$ it reads $v_{n+1} = 2v_n$, which has order $1$.\nFor a linear equation $a_k(n) u_{n+k} + \\dots + a_0(n) u_n = b(n)$ the order is $k$ when neither $a_k$ nor $a_0$ is the zero sequence.\n",
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