Form factor expansions in the 2D Ising model and Painlev\'e VI

We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $\lambda$ - extended diagonal correlation functions $C(N,N;\lambda)$ of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlev\'e VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their $\lambda$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm}(0,1;\lambda)$ 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 $\l$-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 $\l$-extended case, and facilitate other developments.