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Proof: Recall that ¦m (u; s; k+ ) are all interrelated by continuation across
the mth cut. Therefore, ¦m (u; s; k) c = hol(u ’ um ) implies the same
with k + instead of k. Consequently, owing to Theorem 47, we obtain
Fj — (k),m (z) Gm (z) (I ’ e’2πi(sI+Lm ) ) c ≡ 0. Since Fj — (k),m (z) has linearly
independent columns, this implies c = 0.
Examples show that invertibility of I ’ exp[’2πi(sI + Lm )] is not nec-
essary for the linear independence of the columns of ¦m (u; s; k). Whether
or not they can ever be linearly dependent seems to be unknown, but will
not be of importance here.
The above lemma guarantees that the matrices Cnm , for every k ∈ Z and
n = m, 1 ¤ n, m ¤ µ, are determined by (9.12), or even by the identities in
Theorem 46. Exercise 4 on p. 142 shows that from the matrices Ck we can
152 9. Stokes™ Phenomenon

compute the Stokes multipliers V , for j(k) ’ n1 + 1 ¤ ¤ j(k) + n1 . Doing
so for r consecutive values of k then gives enough multipliers to compute
all others with help of (9.4). Hence, in principle the problem of computing
the Stokes multipliers of highest level has been solved. In special situations,
however, there are more e¬ective formulas for this computation.We shall
brie¬‚y illustrate this in the case of Poincar´ rank r = 1 and the leading
term having all distinct eigenvalues. In this situation, the following holds:

• The notions of HLFFS and FFS coincide, since there is only one level
to consider.

• There are ν distinct values un , equal to the negative of the eigenvalues
of the leading term A0 .

• Owing to r = 1 and the form of a FFS, as stated in Exercise 4 on
p. 45, the associated functions here are vectors given by convergent

power series ¦m (u; s; k) = 0 Fn,m (u’um )n’ m ’s /“(1+n’ m ’s),
with not necessarily distinct complex numbers m and ’d + 2kπ <
arg(u ’ um ) < ’d + 2(k + 1)π; for this, compare Exercise 1.

• The formal monodromy matrices here are scalar, and Lm = m. The
vector F0,m equals the mth unit vector em .

In this situation, Theorem 46 states that for n ≺ m

¦m (u; s; k) = ¦n (u; s; k) (I ’ e’2πi (sI+ n)
)’1 Cnm + hol(u ’ un ).

For Re ( n + s) > 0, a term (u ’ un ) n +s ¦(u) tends to 0 whenever ¦(u)
is holomorphic at un . Hence we may evaluate the scalar blocks Cnm by
¬nding the limit of (u ’ un ) n +s ¦m (u; s; k) when u ’ un . If arg(u ’ un ) is
chosen according to ’d + 2kπ < arg(u ’ un ) < ’d + 2(k + 1)π, this limit
equals, using (B.14) (p. 232):

(k) (k)
en cnm cnm “(s + n) n)
= en .
(1 ’ e’2πi(s+ n ) ) “(1 ’ s ’ 2πi

n +s n +s n)
Setting (u ’ un ) = (un ’ u) eπi(s+ , we ¬nd

cnm “(s + n)
n +s
lim (un ’ u) ¦m (u; s; k) = en ,

with ’d + (2k ’ 1)π < arg(un ’ u) < ’d + (2k + 1)π. A similar formula may
be obtained for m ≺ n as well. In Exercise 2 we shall use this result to ¬nd
explicit values for the Stokes multipliers of the two-dimensional con¬‚uent
hypergeometric system.
9.6 Highest-Level Invariants 153

Exercises: Consider a ¬xed normalized HLFFS of a system (3.1) and a
nonsingular direction d.

1. Assume Gm (z) ≡ Ism , for some m, 1 ¤ m ¤ µ. Show ¦m (u; s; k) =
∞ (n’s)/r
/“(1+(n’s)/r), for u ∈ C d , with the branch
0 Fn,m (u’um )
determined according to ’r d < arg(u’um )’2kπi <
of (u’um )
r ’ d + 2π.

2. For A(z) as in Exercise 2 (a) on p. 58, compute the associated func-
tions in terms of hypergeometric ones, and ¬nd all Stokes multipliers.

3. Consider a con¬‚uent hypergeometric system (2.5) (p. 21), with A
having distinct eigenvalues. Show that the associated functions then
satisfy the hypergeometric system (u I + A) φ = ’(s I + B) φ.

9.6 Highest-Level Invariants
In this section, we shall brie¬‚y discuss the notion of equivalence of sys-
tems of meromorphic ODE. This concept was ¬rst introduced and studied
by Birkho¬ [54] in his attempt to classify such systems with respect to
the behavior of solutions near in¬nity: Throughout, we consider two ν-
dimensional systems

|z| > ρ,
z x = A(z) x, z y = B(z) y, (9.14)

with holomorphic coe¬cient matrices, each having a pole at in¬nity of,
possibly distinct, order rA resp. rB . These two systems are said to be
analytically, resp. meromorphically, equivalent to one another, if there exists
an analytic, resp. meromorphic, transformation T (z) satisfying

zT (z) = A(z) T (z) ’ T (z) B(z), |z| > ρ. (9.15)

Indeed, this is an equivalence relation for meromorphic systems near in¬n-
ity, and solutions of equivalent systems essentially behave alike at in¬nity “
however, note that in case of meromorphic equivalence the behavior agrees
only up to integer powers of z.
Birkho¬ introduced the above notion of equivalence in connection with
the following approach toward analyzing the behavior near in¬nity of solu-
tions of systems (3.1). Imagine that we have succeeded in completing the
following two tasks:

• Find a collection of objects, named analytic resp. meromorphic in-
variants, which can in some sense be computed in terms of an arbi-
trarily given system (3.1). These invariants should be such that for
154 9. Stokes™ Phenomenon

each two systems that are analytically resp. meromorphically equiva-
lent all these objects agree, which explains their name. Moreover, the
collection should be complete in the sense that any two systems shar-
ing the same invariants are indeed analytically, resp. meromorphically
equivalent. In other words, the system of analytic resp. meromorphic
invariants characterizes the corresponding equivalence class of sys-
tems (3.1).
• Within each equivalence class of systems with respect to analytic
resp. meromorphic equivalence, ¬nd a unique representative that, in
some sense or another, is the simplest system in this class, and study
the behavior of its solutions near in¬nity.

Assuming that the above has been done, one can then completely anal-
yse the behavior of solutions of an arbitrary system by ¬rst computing
its invariants, thus, determining the equivalence class to which the sys-
tem belongs, and then identifying the corresponding representative for this
equivalence class “ then, the solution of the given system behave as the
ones for the corresponding representative, and their behavior is known!
While Birkho¬ himself found a complete system of invariants only un-
der some restrictive assumptions, the general case has been treated much
later by Balser, Jurkat, and Lutz [33“36, 41]. Also compare Sibuya [250],
or Lutz and Sch¨fke [177]. Here we shall present a simple result on what
we call highest-level meromorphic invariants. To do so, assume that both
systems (9.14) have an essentially irregular singular point at in¬nity. Let
ˆ ˆ
(FA (z), YA (z)) and (FB (z), YB (z)) be HLFFS of the corresponding sys-
tem, with corresponding Stokes™ multipliers Vj,A and Vj,B and formal mon-
odromy matrices LA and LB .
Theorem 48 Let two meromorphic systems (9.14)be given.
(a) Assume the systems to be meromorphically equivalent. Then the data
ˆ ˆ
pairs of the two HLFFS (FA (z), YA (z)) and (FB (z), YB (z)) coincide
up to a renumeration.
ˆ ˆ
(b) Assume that the data pairs of (FA (z), YA (z)) and (FB (z), YB (z)) co-
incide. Then the systems (9.14) are meromorphically equivalent if and
only if there exists a constant invertible matrix D, diagonally blocked
of type (s1 , . . . , sµ ), so that

Vj,A = D’1 Vj,B D, e2πi LA = D’1 e2πi LB D,
j ∈ Z, (9.16)
and in addition YA (z) D’1 YB (z) has moderate growth, for z ’ ∞,
in arbitrary sectors.

Proof: In case of meromorphic equivalence, there exists a meromorphic
transformation T (z) with (9.15). Hence (T (z) FB (z), YB (z)) is an HLFFS
9.7 The Freedom of the Highest-Level Invariants 155

of the ¬rst system in (9.14), which implies (a). In case of identical data
pairs, we conclude from Lemma 12 (p. 142) existence of diagonally blocked
matrices D, constant invertible, and Tq (z), a q-meromorphic transforma-
ˆ ˆ
tion, for which YA (z) = Tq (z) YB (z) D, FA (z) Tq (z) = T (z) FB (z), and
the relation for the Stokes multipliers follow. The ¬rst identity implies
moderate growth of YA (z) D’1 YB (z), while the second one, owing to
properties of k-summability, shows Fj,A (z) Tq (z) = T (z) Fj,B (z) for the
HLNS, for every j ∈ Z. This shows Xj,A (z) = T (z) Xj,B (z) D for ev-
ery j ∈ Z, implying exp[2πi Mj,A ] = D’1 exp[2πi Mj,B ] D. This, together
with (9.5) and the relation between the Stokes multipliers, then implies
exp[2πi LA ] = D’1 exp[2πi LB ] D.
’1 ’1
Conversely, de¬ne T (z) = Xj,A (z) D’1 Xj,B (z) = Fj,A (z) Tq (z) Fj,B (z),
with Tq (z) = YA (z) D’1 YB (z). The relation for the Stokes multipliers
shows that T (z) is independent of j, and from (9.5) and exp[2πi LA ] =
D’1 exp[2πi LB ] D we obtain T (ze2πi ) = T (z). Moreover, moderate growth
of Tq (z), together with the asymptotic of both Fj,A (z) and Fj,B (z), shows
that T (z) is of moderate growth in every sector Sj . Consequently, T (z) can
only have a pole at in¬nity; hence is a meromorphic transformation.
While the Stokes multipliers correspond uniquely to a selected HLFFS
of (3.1), there is a freedom in selecting this HLFFS, and this freedom
exactly re¬‚ects in a change of the Stokes multipliers as in (9.16). Thus, we
may also say as follows: Under the assumptions of the above theorem, the
systems (9.14) are meromorphically equivalent if and only if we can select
corresponding HLFFS of each system so that their Stokes™ multipliers and
formal monodromy factors agree and YA (z) YB (z) is of moderate growth.

1. Show that the notion of analytic, resp. meromorphic, equivalence of
systems is indeed an equivalence relation.

2. For analytic equivalence, verify that the spectrum of the leading term
A0 is invariant. In case of meromorphic equivalence, show that the
same is correct, once the transformation does not change the Poincar´e
rank of the system.

9.7 The Freedom of the Highest-Level Invariants
In Section 9.2 we have shown the Stokes multipliers to satisfy the rela-
tions (9.4). In this section we shall show that, aside from the restriction of
their support, nothing more can be said in general. For this purpose, we
shall prove a technical lemma which is a simpli¬ed version of a result of
156 9. Stokes™ Phenomenon

Sibuya™s [251, Section 6.5]: Consider a closed sector S(ρ) = { z : |z| ≥
ρ, ± ¤ arg z ¤ β }, with ± < β ¬xed. Let X(z) be a ν — ν-matrix, holomor-
phic in the interior of S(ρ) and continuous up to its boundary, and so that
X(z) ¤ c |z|’2 in S(ρ). Finally, de¬ne S — (˜, δ) = { z : |z| > ρ, ± + δ <
¯ ρ ˜
arg z < β + 2π ’ δ }, with δ > 0, and ρ ≥ ρ to be determined.
Lemma 17 Under the above assumptions, for every δ > 0 and su¬ciently
large ρ ≥ ρ, there is a matrix T (z), holomorphic in S — (˜, δ), and tending
˜ ρ
to 0 as z ’ ∞ there, so that:

T (ze2πi ) ’ T (z) = (I + T (z)) X(z), z ∈ S(ρ) © S — (˜, δ).
¯ ρ

Proof: For the proof, we proceed exactly as in [251, pp. 152“162]: For
m ∈ N0 , set Sm = { z : |z| ≥ ρ + δm , ± + δm ¤ arg z ¤ β ’ δm }, Sm =
{ z : |z| ≥ ρ + δm , ± + δm ¤ arg z ¤ β ’ δm + 2π }, with δm = (1 ’ 2’m ) δ.
Beginning with X0 (z) = X(z), de¬ne inductively

’1 du
Um (z) = Xm (u) ,
2πi zm
= Um (z) Xm (z) [I + Um (ze2πi )]’1 ,
Xm+1 (z)

where zm = (˜ + δm ) exp[i(± + β)/2]. Suppose that Xm (z) is holomor-


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