

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 [32]. In this
section, we review Leaver’s solution based on [48
].
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 [19],
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 (21), expressed in terms of a series of
Coulomb wave functions as

