Order conditions for a linear multistep method

A linear multistep method has order $p$ exactly whenfor $0\le q\le p$, with the right side interpreted as zero for $q=0$, and the identity first fails at $q=p+1$.

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.