### 4.3 Outer solution as a series of Coulomb wave functions

The solution as a series of hypergeometric functions discussed in Section 4.2 is convergent at any finite
value of . However, it does not converge at infinity, and hence the asymptotic amplitudes,
and , cannot be determined from it. To determine the asymptotic amplitudes, it is
necessary to construct a solution that is valid at infinity and to match the two solutions in a region
where both solutions converge. The solution convergent at infinity was obtained by Leaver as
a series of Coulomb wave functions [64]. In this section, we review Leaver’s solution based
on [69].
In this section again, by noting the symmetry , we assume without loss of
generality.

First, we define a variable . Let us denote a Teukolsky function by .
We introduce a function by

Then the Teukolsky equation becomes
We see that the right-hand side is explicitly of and the left-hand side is in the form of the Coulomb
wave equation. Therefore, in the limit , we obtain a solution
where is a Coulomb wave function given by
and is the regular confluent hypergeometric function (see [1], Section 13) which is regular at
.
In the same spirit as in Section 4.2, we introduce the renormalized angular momentum .
That is, we add to both sides of Equation (140) to rewrite it as

We denote the formal solution specified by the index by , and expand it in terms of the Coulomb
wave functions as
where . Then, using the recurrence relations among ,
we can derive the recurrence relation among . The result turns out to be identical to the one given by
Equation (123) for . We mention that the extra factor in
Equation (144) is introduced to make the recurrence relation exactly identical to Equation (123).
The fact that we have the same recurrence relation as Equation (123) implies that if we choose the
parameter in Equation (144) to be the same as the one given by a solution of Equation (133) or (136),
the sequence is also the solution for , which is minimal for both . Let us set

By choosing as stated above, we have the asymptotic value for the ratio of two successive terms of
as
Using an asymptotic property of the Coulomb wave functions, we have
We thus find that the series (144) converges at or equivalently .
The fact that we can use the same as in the case of hypergeometric functions to obtain the
convergence of the series of the Coulomb wave functions is crucial to match the horizon and outer
solutions.

Here, we note an analytic property of the confluent hypergeometric function (see [34], Page 259),

where is the irregular confluent hypergeometric function, and is assumed. Using this with
the identities
we can rewrite (for ) as
where
By noting an asymptotic behavior of at large ,
we find
where
We can see that the functions and are incoming-wave and outgoing wave solutions
at infinity, respectively. In particular, we have the upgoing solution, defined for by
the asymptotic behavior (20), expressed in terms of a series of Coulomb wave functions as