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Factorization of Ising correlations C(M, N) for ν = −k and M + N odd, M ⩽ N, T < T c and their lambda extensions

We study the factorizations of Ising low-temperature correlations C(M, N) for ν = −k and M + N odd, M ⩽ N, for both the cases M ≠ 0 where there are two factors, and M = 0 where there are four factors. We find that the two factors for M ≠ 0 satisfy the same non-linear differential equation and, similarly, for M = 0 the four factors each satisfy Okamoto sigma-form of Painlevé VI equations with the same Okamoto parameters. Using a Landen transformation we show, for M ≠ 0, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlevé VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlevé VI equations which generalizes the factorization of the correlations C(M, N) to an additive decomposition of the corresponding sigma’s solutions of the Okamoto sigma-form of Painlevé VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M, N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy–Widom (Painlevé V) relations between the sum and difference of sigma’s to this Painlevé VI framework.