

4.2 Horizon solution in series of hypergeometric
functions
As in Section 3, we focus on the ingoing
wave function of the radial Teukolsky equation (15). Since the analysis below is applicable
to any spin,
,
,
,
, and
, we do not specify it except when it is needed.
Also, the analysis is not restricted to the case
unless so stated
explicitly. For general spin weight
, the homogeneous Teukolsky equation is
given by
As before, taking account of the symmetry
, we may
assume
if necessary.
The Teukolsky equation has two regular
singularities at
, and one irregular singularity at
. This implies
that it cannot be represented in the form of a single
hypergeometric equation. However, if we focus on the solution near
the horizon, it may be approximated by a hypergeometric equation.
This motivates us to consider the solution expressed in terms of a
series of hypergeometric functions.
We define the independent variable
in place of
(
) as
where
For later convenience, we also introduce
and
. Taking into account the structure
of the singularities at
, we put the ingoing wave Teukolsky function
as
Then the radial Teukolsky equation becomes
where a prime denotes
. The left-hand side of Equation (117) is in the form of a hypergeometric
equation. In the limit
, noting Equation (110), we find that a solution that is finite
at
is given
by
For a general value of
, Equation (117) suggests that a solution may be
expanded in a series of hypergeometric functions with
being a kind of
expansion parameter. This idea was extensively developed by
Leaver [32
]. Leaver obtained solutions of
the Teukolsky equation expressed in a series of the Coulomb wave
functions. The MST formalism is an elegant reformulation of the one
by Leaver [32
].
The essential point is to introduce the so-called
renormalized angular momentum
, which is a generalization of
, to a non-integer
value such that the Teukolsky equation admits a solution in a
convergent series of hypergeometric functions. Namely, we add the
term
to both sides of Equation (117) to rewrite it as
Of course, no modification is done to the original equation, and
is just an
irrelevant parameter at this stage. A trick is to consider the
right-hand side of the above equation as a perturbation, and look
for a formal solution specified by the index
in a series expansion
form. Then, only after we obtain the formal solution, we require
that the series should converge, and this requirement determines
the value of
.
Note that, if we take the limit
, we must have
(or
) to
assure
and to
recover the solution (118).
Let us denote the formal solution specified by a
value of
by
. We express it in
the series form,
Here, the hypergeometric functions
satisfy the recurrence
relations [33
],
Inserting the series (120) into Equation (119) and using the above recurrence
relations, we obtain a three-term recurrence relation among the
expansion coefficients
. It is given by
where
The convergence of the series (120) is determined by the asymptotic
behaviors of the coefficients
at
. We thus discuss properties of the three-term
recurrence relation (123) and the role of the parameter
in detail.
The general solution of the recurrence
relation (123) is expressed in terms of two linearly
independent solutions
and
(
,
). According to the theory of three-term recurrence
relations (see [26
], Page 31) when there
exists a pair of solutions that satisfy
then the solution
is called minimal as
(
). Any
non-minimal solution is called dominant. The minimal solution (either as
or as
) is
determined uniquely up to an overall normalization factor.
The three-term recurrence relation is closely
related to continued fractions. We introduce
We can express
and
in terms of
continued fractions as
These expressions for
and
are valid if the respective continued fractions
converge. It is proved (see [26
], Page 31) that the
continued fraction (127) converges if and only if the recurrence
relation (123) possesses a minimal solution as
, and the
same for the continued fraction (128) as
.
Analysis of the asymptotic behavior of (123) shows that, as long as
is finite, there
exists a set of two independent solutions that behave as (see,
e.g., [26],
Page 35)
and another set of two independent solutions that behave as
Thus,
is minimal as
and
is minimal as
.
Since the recurrence relation (123) possesses minimal solutions as
, the
continued fractions on the right-hand sides of Equations (127) and (128) converge for
and
. In general, however,
and
do not coincide.
Here, we use the freedom of
to obtain a consistent solution. Let
be a sequence
that is minimal for both
. We then have expressions for
and
in
terms of continued fractions as
This implies
Thus, if we choose
such that it satisfies the implicit equation for
,
Equation (133), for a certain
, we obtain a unique
minimal solution
that is valid over the entire range of
,
,
that is
Note that if Equation (133) for a certain value of
is satisfied, it is
automatically satisfied for any other value of
.
The minimal solution is also important for the
convergence of the series (120). For the minimal solution
, together with
the properties of the hypergeometric functions
for large
, we find
Thus, the series of hypergeometric functions (120) converges for all
in the range
(in
fact, for all complex values of
except at
), provided that the coefficients are given by the
minimal solution.
Instead of Equation (133), we may consider an equivalent but
practically more convenient form of an equation that determines the
value of
.
Dividing Equation (123) by
, we find
where
and
are those
given by the continued fractions (131) and (132), respectively. Although the value of
in this equation
is arbitrary, it is convenient to set
to solve for
.
For later use, we need a series expression for
with better
convergence properties at large
. Using analytic properties of
hypergeometric functions, we have
where
This expression explicitly exhibits the symmetry of
under the
interchange of
and
. This
is a result of the fact that
is invariant under the interchange
. Accordingly, the recurrence relation (123) has the structure that
satisfies the
same recurrence relation as
.
Finally, we note that if
is a solution of
Equation (133) or (136),
with an arbitrary integer
is also a solution,
since
appears
only in the combination of
. Thus, Equation (133) or (136) contains an infinite number of roots.
However, not all of these can be used to express a solution we
want. As noted in the earlier part of this section, in order to
reproduce the solution in the limit
, Equation (118), we must have
(or
by
symmetry). Thus, we impose a constraint on
such that it must
continuously approach
as
.

