Exact expressions for the 5-Point Liouville conformal block with a level-two degenerate field insertion

In this paper we investigate 5-point Liouville conformal block with a level 2 degenerate field insertion. Our main tool is the BPZ differential equation, which, upon placing three of the insertions at the standard positions $\mathrm{\infty }$, $1$, and $0$, reduces to a linear differential equation which is of order two in the degenerate insertion point $z$, and order one in the remaining point $x$. In a previous paper, it was conjectured that the solution could be expressed in terms of a single hypergeometric function and its derivative, with coefficients computable via recursive relations up to the desired order $x^k$. In this paper, we simplify these recursion relations and provide a rigorous inductive proof of the conjecture. Our representation of the 5-point conformal block readily facilitates the connection between various analyticity regions through classical connection formulae for the hypergeometric function. In the quasi-classical limit, the 5-point BPZ equation reduces to the Heun equation. Consequently, we recover a recently proposed representation of the Heun equation in terms of a single hypergeometric function, which has proven to be highly effective in the analysis of gravitational perturbation of black holes.