Form factor expansions in the 2D Ising model and Painlevé VI

We derive a Toda-type recurrence relation, in both high- and low-temperature regimes, for the λ  -extended diagonal correlation functions C(N,N;λ)C(N,N;λ) of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their λ-extended counterparts.We also conjecture a closed form expression for the simplest off-diagonal case C±(0,1;λ)C±(0,1;λ) where a connection to PVI theory is not known. Combined with the results for diagonal correlations these give all the initial conditions required for the λ-extended version of quadratic difference equations for the correlation functions discovered by McCoy, Perk and Wu. The results obtained here should provide a further potential algorithmic improvement in the λ-extended case, and facilitate other developments.