(untitled) (Journal of Computational Mathematics, 2018)
Using least parameters, we expand the step-transition operator of any linear multi-step method (LMSM) up to O(τ ^s+5) with order s=1 and rewrite the expansion of the step-transition operator for s=2 (obtained by the second author in a former paper). We prove that in the conjugate relation G_3^λτ ° G_1^τ=G_2^τ° G_3^λτ with G_1 being an LMSM, (1) the order of G_2 can not be higher than that of G_1; (2) if G_3 is also an LMSM and G_2 is a symplectic B-series, then the orders of G_1, G_2 and G_3 must be 2, 2 and 1 respectively.