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We study the factorizations of Ising low-temperature correlations C(M,N) for$\nu=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there aretwo factors, and $M=0$ where there are four factors. We find that the twofactors for $ M \neq 0$ satisfy the same non-linear differential equation and,similarly, for M=0 the four factors each satisfy Okamoto sigma-form ofPainlev\'e VI equations with the same Okamoto parameters. Using a Landentransformation we show, for $M\neq 0$, that the previous non-lineardifferential equation can actually be reduced to an Okamoto sigma-form ofPainlev\'e VI equation. For both the two and four factor case, we find thatthere is a one parameter family of boundary conditions on the Okamotosigma-form of Painlev\'e VI equations which generalizes the factorization ofthe correlations C(M,N) to an additive decomposition of the correspondingsigma's solutions of the Okamoto sigma-form of Painlev\'e VI equation which wecall lambda extensions. At a special value of the parameter, thelambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials inthe complete elliptic functions of the first and second kind. We alsogeneralize some Tracy-Widom (Painlev\'e V) relations between the sum anddifference of sigma's to this Painlev\'e VI framework.

Factorization of Ising correlations C(M,N) for $ \nu= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions