Remarks on the convergence of the β-expansion of and s=1 Ising models

In a recent work (J. Math. Phys. 43 (2002) 1390) we derived analytical expressions for the coefficients in the high temperature expansion of the Helmholtz free energy of periodic one-dimensional chains in the cumulant method, for arbitrary order in β. In order to do so, an auxiliary function was defined. The purpose of the present paper is to show that a poor convergence of the high temperature expansions of thermodynamic quantities are signaled by the existence of non-physical singularity(ies) in the auxiliary function. Our discussion is based on the exactly solvable spin-1 and spin- Ising models, whose respective auxiliary functions have algebraic expressions valid for β [0,∞).