= Exponential-symbol order criterion for a multistep method
{title2=$\rho(e^z)-z\sigma(e^z)=O(z^{p+1})$}
The <Taylor expansion> of the exact-solution residual for a <linear multistep method> has the same coefficient sequence as
$$
\rho(e^z)-z\sigma(e^z).
$$
Vanishing through degree $p$ is therefore equivalent to order at least $p$. Exact order $p$ additionally requires a nonzero coefficient at degree $p+1$. This symbol conveniently collects all polynomial consistency and order conditions.
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