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S(z) converges as desired. To do this, we proceed similarly to the proof of
Lemma 1 (p. 2): We have An ¤ c K n for every constant K > 1/ρ and
su¬ciently large c > 0, depending upon K. Abbreviating the right-hand
n’1
side of (2.3) by Bn , we obtain Bn ¤ c m=0 K n’m Sm , n ≥ 1. Divide
(2.3) by n and think of the elements of Sn arranged, in one way or another,
into a vector of length ν 2 . Doing so, we obtain a linear system of equa-
tions with a coe¬cient matrix of size ν 2 — ν 2 , whose entries are bounded
functions of n. Its determinant tends to 1 as n ’ ∞, and is never going
to vanish, according to E. Consequently, the inverse of the coe¬cient ma-
trix also is a bounded function of n. These observations imply the estimate
Sn ¤ n’1 c Bn , n ≥ 1, with su¬ciently large c, independent of n. Let
˜ ˜
’1 n’1
sm , n ≥ 1, with c = c c, and conclude
n’m
s0 = S0 , sn = n c m=0 K
ˆ ˆ˜

by induction Sn ¤ sn , n ≥ 0. The power series f (z) = 0 sn z n for-
mally satis¬es f (z) = cK f (z) (1 ’ Kz)’1 , and as in the proof of Lemma 1
ˆ
(p. 2) we obtain convergence of f (z) for |z| < K ’1 , hence convergence of
S(z) for |z| < ρ. 2
Note that the above theorem coincides with Lemma 1 (p. 2) in case of
A0 = 0, which trivially has good spectrum. Moreover, the theorem obvi-
ously solves the three problems stated above in quite a satisfactory manner:
The computation of a monodromy matrix is trivial, the corresponding S(z)
has a removable singularity at the origin, det S(z) attains value 1 there, and
the coe¬cients of its Laurent expansion, which here is a power series, can
20 2. Singularities of First Kind

be recursively computed from (2.3). As we shall see, the situation gets
more complicated for systems with general spectrum: First of all, A0 will
no longer be a monodromy matrix, although closely related to one, and
secondly the single-valued part of fundamental solutions has a somewhat
more complicated structure as well. Nonetheless, we shall also be able to
completely analyze the structure of fundamental solutions in the general
situation.

Exercises: In the following exercises, consider a ¬xed system (2.1) with
good spectrum.
1. Give a di¬erent proof for the existence part of Theorem 5 as follows:
For N ∈ N, assume that we computed S1 , . . . , SN from (2.3), and let
N
PN (z) = I + 1 Sn z n , B(z) = A(z) PN (z) ’ zPN (z) ’ PN (z) A0 ,
˜
X(z) = X(z) ’ PN (z) z A0 . For su¬ciently large N , show that X(z)
solves (2.1) if and only if
z
du
˜ ˜
B(u) uA0 + A(u) X(u)
X(z) = . (2.4)
u
0

Then, by the standard iteration method, show that (2.4) has a so-
˜ ˜ ˜
lution X(z) = S(z) z A0 , with S(z) holomorphic near the origin and
vanishing of order at least N + 1.
2. Let T be invertible, so that J = T ’1 A0 T is in Jordan canonical
form. Show that (2.1) has a fundamental solution X(z) = S(z) z J ,

with S(z) = T + n=1 Sn z n , |z| < ρ.
3. Let s0 be an eigenvector of A0 , corresponding to the eigenvalue µ.
Show that (2.1) has a solution x(z) = s(z) z µ , with s(z) = s0 +

n=1 sn z , |z| < ρ. Such solutions are called Floquet solutions, and
n

we refer to µ as the corresponding Floquet exponent. Find the recur-
sion formulas for the coe¬cients sn .
4. Show that (2.1) has k linearly independent Floquet solutions if and
only if A0 has k linearly independent eigenvectors. In particular, (2.1)
has a fundamental solution consisting of Floquet solutions if and only
if A0 is diagonalizable.
5. In dimension ν = 2, let
µ0
µ ∈ C.
A0 = ,

Show that (2.1) has a fundamental solution consisting of one Floquet
solution and another one of the form x(z) = (s1 (z) + s2 (z) log z) z µ ,

sj (z) = 0 sn z n , |z| < ρ. Try to generalize this to higher dimen-
sions.
2.2 Con¬‚uent Hypergeometric Systems 21

2.2 Con¬‚uent Hypergeometric Systems
As an application of the results of Section 2.1, we study in more detail the
very special case of

A, B ∈ C ν—ν .
zx = (zA + B) x, (2.5)

We shall refer to this case as the con¬‚uent hypergeometric system, since
it may be considered as a generalization of the second-order scalar ODE
bearing the same name, introduced in Exercise 3. Under various addi-
tional assumptions on A and B, such systems, and/or the closely related
hypergeometric systems that we shall look at in the next section, have
been studied, e.g., by Jurkat, Lutz, and Peyerimho¬ [147, 148], Okubo and
Takano [207], Balser, Jurkat, and Lutz [37, 41], Kohno and Yokoyama [161],
Balser [11“13, 20], Sch¨fke [240], Okubo, Takano, and Yoshida [208], and
a
Yokoyama [288, 289].
For simplicity we shall here restrict our discussion to the case where B
is diagonalizable and E holds. So according to Exercise 4 on p. 20 we have

ν linearly independent Floquet solutions x(z) = n=0 sn z n+µ , where µ
is an eigenvalue of B and s0 a corresponding eigenvector, and the series
converges for every z ∈ C . The coe¬cients satisfy the following simple
recursion relation:

sn = ((n + µ)I ’ B)’1 A sn’1 , n ≥ 1. (2.6)

Note that the inverse matrix always exists according to E. Hence we see
that sn is a product of ¬nitely many matrices times s0 . To further sim-
plify (2.6), we may even assume that B is, indeed, a diagonal matrix
D = diag [µ1 , . . . , µν ], since otherwise we have B = T DT ’1 for some in-
vertible T , and setting A = T AT ’1 , sn = T sn , this leads to a similar
˜ ˜
recursion for sn . Then, µ is one of the values µk and s0 a corresponding
˜
unit vector.
Despite of the relatively simple form of (2.6), we will have to make some
severe restrictions before we succeed in computing sn in closed form. Es-
sentially, there are two cases that we shall now present.
To begin, consider (2.6) in the special case of

ab µ1 0
A= , B= , (2.7)
cd 0 µ2

assuming that µ1 ’ µ2 = Z except for µ1 = µ2 , so that E holds. In this
case, let us try to explicitly compute the Floquet solution corresponding to
the exponent µ = µ1 ; the computation of the other one follows the same
lines. Denoting the two coordinates of sn by fn , gn , we ¬nd that (2.6) is
equivalent to

n ≥ 1,
nfn = afn’1 + bgn’1 , (n + β)gn = cfn’1 + dgn’1 ,
22 2. Singularities of First Kind

for β = µ1 ’ µ2 , and the initial conditions f0 = 1, g0 = 0. Note n + β =
0, n ≥ 1, according to E. This implies

(n + 1)(n + β)fn+1 = (n + β)(afn + bgn )
= a(n + β)fn + b(cfn’1 + dgn’1 ).

Using the original relations, we can eliminate gn’1 to obtain the following
second order recursion for the sequence (fn ):

(n + 1)(n + β)fn+1 = [n(a + d) + aβ]fn ’ (ad ’ bc)fn’1 , n ≥ 1,

together with the initial conditions f0 = 1, f1 = a. Unfortunately, such a
recursion in general is still very di¬cult to solve “ however, if ad’bc = det A
would vanish, this would reduce to a ¬rst-order relation. Luckily, there is
a little trick to achieve this: Substitute 1 x = e»z x into the system (2.1) to
˜
obtain the equivalent system z x = (A(z) ’ z») x. In case of a con¬‚uent
˜ ˜
hypergeometric system and » equal to an eigenvalue of A, we arrive at
another such system with det A = 0. Note that if we computed a Floquet
solution of the new system, we then can reverse the transformation to
obtain such a solution for the original one.
To proceed, let us now assume ad ’ bc = 0; hence one eigenvalue of
A vanishes. Then » = a + d is equal to the second, possibly nonzero,
eigenvalue, and the above recursion becomes

n ≥ 1,
(n + 1)(n + β)fn+1 = (n» + aβ)fn ,

with f1 = a. Leaving the case of » = 0 as an exercise and writing ± = aβ/»
for » = 0, we ¬nd 2

n
(±)n
fn = » (β)n 
n!
, n ≥ 1.
»n’1 (±+1)n’1 
gn = c (n’1)! (β+1)n

Thus, one can explicitly express the Floquet solution of the con¬‚uent hy-
pergeometric system in terms of the following well-known higher transcen-
dental function:

Confluent Hypergeometric Function
For ± ∈ C , β ∈ C \ {0, ’1, ’2, . . .}, the function

(±)n n
z ∈ C,
F (±; β; z) = z,
n! (β)n
n=0


1 Note
that what is done here is a trivial case of what will be introduced as analytic
transformations in the following section.
2 Here we use the Pochhammer symbol (±) = 1, (±) = ± · . . . · (± + n ’ 1), n ≥ 1.
0 n
2.2 Con¬‚uent Hypergeometric Systems 23

is called con¬‚uent hypergeometric function. Another name for
this function is Kummer™s function. It arises in solutions of
the con¬‚uent hypergeometric di¬erential equation introduced in
the exercises below. For ± = ’m, m ∈ N0 , the function is a
polynomial of degree m; otherwise, it is an entire function of
exponential order 1 and ¬nite type.
In the case of » = 0, the coe¬cients fn obviously decrease at a much faster
rate. This is why the corresponding functions are of smaller exponential
order. In a way, it is typical in the theory of linear systems of meromorphic
ODE to have a “generic situation” (here: » = 0 and ± = 0, ’1, . . .) in
which solutions show a certain behavior (here, they are entire functions of
exponential order 1 and ¬nite type), while in the remaining case they are
essentially di¬erent (of smaller order, or even polynomials). To explicitly
¬nd the solutions in these exceptional cases, we de¬ne another type of
special functions, which are very important in applications:
Bessel™s Function
For µ ∈ C , the function

(’1)n
(z/2)2n+µ , z ∈ C,
Jµ (z) =
n! “(1 + µ + n)
n=0

is called Bessel™s function. Removing the power (z/2)µ , we ob-
tain an entire function of exponential order 1 and ¬nite type.
The function is a solution of a scalar second-order ODE, called
Bessel™s di¬erential equation.
In Exercise 2 we shall show that Bessel™s function also arises in solutions
of (2.6) for ν = 2 and nilpotent A, i.e., » = 0.
Next, we brie¬‚y mention another special case of (2.5) where Floquet
solutions can be computed in closed form: For arbitrary dimension ν, let
® 
»1 0 . . . 0 a1
 0 »2 . . . a2 
0
 
. . .
. .. .
A = diag [0, . . . , 0, 1], B =  . .
. . (2.8)
.
. . . .
°0 0 . . . »ν’1 aν’1 »
b1 b2 . . . bν’1 »ν
Under certain generic additional assumptions one can explicitly compute
Floquet solutions of (2.5), using the following well-known functions:
Generalized Confluent Hypergeometric Functions
For m ≥ 1, ±j ∈ C , βj ∈ C \ {0, ’1, ’2, . . .}, 1 ¤ j ¤ m, the
function

(±1 )n · . . . · (±m )n n
F (±1 , . . . , ±m ; β1 , . . . , βm ; z) = z
n! (β1 )n · . . . · (βm )n
n=0
24 2. Singularities of First Kind

(with radius of convergence of the series equal to in¬nity), is
called generalized con¬‚uent hypergeometric function. For some
parameter values (which?), the function is a polynomial, and
otherwise an entire function of exponential order 1 and ¬nite
type. The function arises in solutions of a scalar (m + 1)st-
order ODE, called generalized con¬‚uent hypergeometric di¬er-
ential equation, introduced in the exercises below.

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