Differential equations for intertwining operators among untwisted and twisted modules

Given any vertex operator algebra $V$ with an automorphism $g$, we derive a Jacobi identity for an intertwining operator $\mathcal{Y}$ of type $\left( \begin{smallmatrix} W_3\\ W_1 \, W_2 \end{smallmatrix}\right)$ when $W_1$ is an untwisted $V$-module, and $W_2$ and $W_3$ are $g$-twisted $V$-modules. We say such an intertwining operator is of $\left(\!\begin{smallmatrix} g\\ 1 \ g \end{smallmatrix}\!\right)$-type. Using the Jacobi identity, we obtain homogeneous linear differential equations satisfied by the multi-series $\langle w_0, \mathcal{Y}_1(w_1,z_1) \cdots \mathcal{Y}_N(w_N,z_N) w_{N+1} \rangle$ when $\mathcal{Y}_j$ are of $\left(\!\begin{smallmatrix} g\\ 1 \ g \end{smallmatrix}\!\right)$-type and the modules are $C_1$-cofinite and discretely graded. In the special case that $V$ is an affine vertex operator algebra, we derive the ``twisted KZ equations" and show that its solutions have regular singularities at certain prescribed points when $g$ has finite order. When $V$ is general and $g$ has finite order, we use the theory of regular singular points to prove that the multi-series $\langle w_0, \mathcal{Y}_1(w_1,z_1) \cdots \mathcal{Y}_N(w_N,z_N) w_{N+1} \rangle$ converges absolutely to a multivalued analytic function when $|z_1| > \cdots > |z_N| > 0$ and analytically extends to the region $z_i, z_i - z_j \neq 0$. Furthermore, when $N = 2$, we show that these multivalued functions have regular singularities at certain prescribed points.