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  "body": "\n## Definition\n\nLet $K$ be a field and let $d \\ge 1$ be an integer.\nLet $a_0, \\dots, a_{d-1} \\colon \\N \\to K$ and $g \\colon \\N \\to K$ be [[sequences|0G9EKM0E15]].\nA linear recurrence of order $d$ is the [[recurrence relation|07TZV87Y0T]]\n$$u_{n+d} + a_{d-1}(n)\\, u_{n+d-1} + \\cdots + a_0(n)\\, u_n = g(n) \\qquad (n \\in \\N)$$\nfor a sequence $u \\colon \\N \\to K$.\nThe sequence $g$ is its right-hand side.\n\n1. The recurrence is *homogeneous* if $g(n) = 0$ for all $n$.\n2. The recurrence is *inhomogeneous* if $g(n) \\ne 0$ for some $n$.\n3. The *associated homogeneous recurrence* is the one with the same $a_0, \\dots, a_{d-1}$ and with $g$ replaced by $0$.\n\n## Remarks\n\nExample: $u_{n+1} - 2 u_n = 1$ is inhomogeneous of order $1$, with $a_0(n) = -2$ and $g(n) = 1$.\nIts associated homogeneous recurrence is $u_{n+1} - 2 u_n = 0$, whose solutions are $c\\, 2^n$.\nNon-example: $u_{n+1} = u_n^2 + 1$ is not a linear recurrence at all, since the term $u_n^2$ is not linear in $u_n$; so it is neither homogeneous nor inhomogeneous in this sense.\nWhen all $a_i$ are constant, the homogeneous case is a [[linear recurrence with constant coefficients|081THG13JZ]].\nOther names are homogeneous and nonhomogeneous linear difference equations.\nThe difference of two solutions of the inhomogeneous recurrence solves the associated homogeneous one.\n",
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