Exponential-symbol order criterion for a multistep method ($\rho (e^z)-z\sigma (e^z)=O(z^{p+1})$)

The Taylor expansion of the exact-solution residual for a linear multistep method has the same coefficient sequence asVanishing 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.