Differential equations for Dotsenko--Fateev integrals: the case of degenerate fields ${\mathrm{\Phi }}_{n,1}$

We study higher-order differential equations corresponding to integrals of the Dotsenko–Fateev type and associated with four-point correlation functions with degenerate fields ${\mathrm{\Phi }}_{n,1}$. We determine the general structure and coefficients of the equations in terms of the integral parameters $\{A,B,C,g\}$. In particular, special cases of these equations in CFT are used for verification. In addition, the relationship between higher-order differential equations and hypergeometric operators is established.