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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $k \\ge 1$ be an integer.\nLet $a_0, a_1, \\dots, a_k$ and $b$ be sequences in $K$, indexed by $n \\in \\N$, with $a_k(n) \\ne 0$ for all $n$.\nThe *linear difference equation* with these coefficients is the equation\n$$a_k(n)\\, u_{n+k} + a_{k-1}(n)\\, u_{n+k-1} + \\dots + a_0(n)\\, u_n = b(n) \\qquad (n \\ge 0)$$\nfor an unknown sequence $(u_n)_{n \\ge 0}$ in $K$.\nIt is *homogeneous* if $b(n) = 0$ for all $n$, and *inhomogeneous* otherwise.\nIt has *constant coefficients* if each $a_i$ is a constant sequence.\n\n## Remarks\n\nSince $a_k(n) \\ne 0$, the equation can be solved for $u_{n+k}$, so it is a [[recurrence relation|07TZV87Y0T]] of order at most $k$.\nExample: $u_{n+1} - (n+1) u_n = 0$ is a homogeneous linear difference equation with $k = 1$, $a_1 = 1$ and $a_0(n) = -(n+1)$; with $u_0 = 1$ its solution is $u_n = n!$.\nNon-example: $u_{n+1} = u_n^2$ is not linear, because the unknown appears squared; the sequences $u = (1, 1, 1, \\dots)$ and $2u = (2, 2, 2, \\dots)$ show that multiples of solutions need not be solutions.\nIn terms of the [[shift operator|91QMF4R1CY]] $E$, the equation reads $L u = b$ with $L = a_k E^k + \\dots + a_0$, a linear operator on sequences.\nThe constant coefficient case is the [[linear recurrence with constant coefficients|081THG13JZ]].\n",
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