### 3.2 Horizon solution;

In this section, we first consider the solution near the horizon, which we call the horizon solution, based on [85]. To do so, we assume and treat as a small number, but leave the ratio arbitrary. We change the independent variable to and the wave function to
Note that the horizon corresponds to . We then have
where a prime denotes differentiation with respect to . We look for a solution which is regular at .

First, we consider the lowest order solution by setting in Equation (74). The boundary condition (72) requires that at . The solution that satisfies the boundary condition is

Thus, the lowest order solution is a polynomial of order in .

Next, we consider the solution accurate to . We neglect the terms of in Equation (74). Then, the wave equation takes the form of a hypergeometric equation,

with parameters
The two linearly independent solutions are and , where is the hypergeometric function. However, only the first solution is regular at . Therefore, we obtain

The above solution must be matched with the solution obtained from the post-Minkowski expansion of Equation (70), which we call the outer solution, in a region where both solutions are valid. It is the region where the post-Newtonian expansion is applied, i.e., the region . For this purpose, we rewrite Equation (78) as (see, e.g., Equation (15.3.8) of [1])

This naturally allows the expansion in . It should be noted that the second term in the square brackets of the above expression is meaningless as it is, since the factor diverges for integer . So, when evaluating the second term, we first have to extend to a non-integer number. Then, only after expanding it in terms of , we should take the limit of an integer . One then finds that this procedure gives rise to an additional factor of . For , it therefore becomes higher in than the first term. Then, we obtain
where
and is the digamma function,
and is the Euler constant.

As we will see below, the above solution is accurate enough to determine the boundary condition of the outer solution up to the 6PN order of expansion.