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

2

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.

(k)

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

e

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 ).

(k)

For Re ( n + s) > 0, a term (u ’ un ) n +s ¦(u) tends to 0 whenever ¦(u)

(k)

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)

eπi(s+

= en .

(1 ’ e’2πi(s+ n ) ) “(1 ’ s ’ 2πi

n)

n +s n +s n)

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

(k)

cnm “(s + n)

n +s

lim (un ’ u) ¦m (u; s; k) = en ,

2πi

u’un

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 )

’s/r

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

a

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)

’1

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

ˆ

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

’1

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),

’1

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

2

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

’1

formal monodromy factors agree and YA (z) YB (z) is of moderate growth.

Exercises:

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) ,

u’z

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-