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  "body": "\n## Statement\n\nLet $A$ be the advancement operator on sequences, so that $(Af)(n) = f(n+1)$. Let $r_1, \\dots, r_k$ be distinct non-zero constants. Consider the equation $(A - r_1)(A - r_2)\\cdots(A - r_k) f = 0$.\n\nThen every [[solution|0ZV2YKKZ58]] $f$ of this equation can be written as $f(n) = c_1 r_1^n + c_2 r_2^n + \\dots + c_k r_k^n$ for some constants $c_1, \\dots, c_k$.\n\n## Remarks\n\nThe polynomial $p(A) = (A - r_1)\\cdots(A - r_k)$ is the [[characteristic polynomial|0BWDJGQYST]] of the recurrence, and $r_1, \\dots, r_k$ are its roots. The statement says that the sequences $r_i^n$ span all solutions.\n",
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        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_recurrence_rigorous.ptx#L103"
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  "msc": [
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  "name": "Solutions of an advancement operator equation with distinct roots",
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