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 = ,

1µ

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.