On general-n coefficients in series expansions for row spin-spin correlation functions in the two-dimensional Ising model

We consider spin-spin correlation functions for spins along a row, R _n = ⟨σ _0,0 σ _n,0⟩, in the two-dimensional Ising model. We discuss a method for calculating general-n expressions for coefficients in high-temperature and low-temperature series expansions of R _n and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for a nonlinear ordinary differential equation whose solution would determine R _n , analogous to the known nonlinear ordinary differential equation whose solution determines the diagonal correlation function ⟨σ _0,0 σ _n,n ⟩ in this model.