Exercises: In the ¬rst two of the following exercises, consider a ¬xed

con¬‚uent hypergeometric system with A, B as in (2.7). Also, restrict µ1 ’µ2

so that E holds. For the last one, consider m ∈ N and parameters ±j , βj as

in the de¬nition of the generalized con¬‚uent hypergeometric function, and

let δ denote the di¬erential operator z(d/dz).

1. For A having distinct eigenvalues, but not assuming det A = 0, ex-

plicitly express one Floquet solution of (2.5) using exponential and

con¬‚uent hypergeometric functions.

2. For A having equal eigenvalues, do the same as above in terms of

exponential and Bessel functions.

3. Verify that the (scalar) con¬‚uent hypergeometric ODE

z y ’ (z ’ β)y ’ ±y = 0

for β = 0, ’1, ’2, . . . has the solution F (±; β; z). Moreover, verify

that the equation has only one solution, aside from a constant factor,

which is holomorphic at the origin. Note that in the literature one

can ¬nd another second-order ODE, closely related to the one above,

bearing the same name.

4. Show Kummer™s transformation

F (±; β; z) = ez F (β ’ ±; β; ’z), z ∈ C , β = 0, ’1, ’2, . . .

5. For Re β > Re ± > 0, show

1

“(β)

ezt t±’1 (1 ’ t)β’±’1 dt.

F (±; β; z) =

“(±) “(β ’ ±) 0

6. Verify that Bessel™s equation

z 2 y + zy + (z 2 ’ µ2 )y = 0

has the solution Jµ (z).

2.3 Hypergeometric Systems 25

7. Show that F (±1 , . . . , ±m ; β1 , . . . , βm ; z) satis¬es the following scalar

ODE of (m + 1)st order, called the con¬‚uent hypergeometric di¬er-

ential equation:

d

[(δ + ±1 ) · . . . · (δ + ±m ) ’ (δ + β1 ) · . . . · (δ + βm ) ] y = 0.

dz

2.3 Hypergeometric Systems

In this section we consider so-called hypergeometric systems, which are of

the form

(A ’ zI) x = B x, A, B ∈ C ν—ν . (2.9)

As shall become clear in the following chapters, (2.9) is intimately related,

via Laplace transform, to a con¬‚uent hypergeometric system with slightly

di¬erent A and B. Another relation is through a process called con¬‚uence,

which shall not be discussed here but is mentioned to explain the name for

the systems (2.5).

For simplicity, we restrict our discussion to A = diag [»1 , . . . , »ν ], with

not necessarily distinct values »k . The system (2.9) then is meromorphic in

C with ¬rst-order poles at the numbers »1 , . . . , »ν . If all the »k are equal, a

change of variable z = u+», reduces the system to one treated in Exercise 1

on p. 9, so we exclude this case here. For »1 = . . . = »ν’1 = »ν , we shall

compute Floquet solutions of (2.9) using the hypergeometric function resp.

its generalized versions. For ν = 3 and three distinct values »k , the system,

under some additional assumptions, should be closely related to Heun™s

ODE, and the corresponding con¬‚uent one to one of its various con¬‚uent

forms [234], but to the author™s knowledge these relations have never been

worked out. We leave this to the reader as a research problem.

Making a change of variable z = au + b, a = 0, one can achieve »1 = 0,

»ν = 1. This shows that in dimension ν = 2 we may assume

0 0 ab

A= , B= .

0 1 cd

Putting the system into the form (2.1), we ¬nd that the Floquet exponents

are ’a and 0. For simplicity we restrict to a noninteger value for a, so

that in particular E is satis¬ed. Using the same notation as in the previous

section, the recursion relations for the coe¬cients of a Floquet solution

with exponent µ = ’a are

0 = nfn + b gn , (n + 1 ’ a)gn+1 = c fn + (n + d ’ a)gn , n ≥ 0,

26 2. Singularities of First Kind

with f0 = 1, g0 = 0. Eliminating fn , we obtain n(n + 1 ’ a) gn+1 =

(n + ±)(n + β) gn , n ≥ 1, with ± + β = d ’ a, ±β = ’cb and the initial

condition g1 = c/(1 ’ a). With γ = 1 ’ a, this implies

(±)n (β)n

fn = n! (γ)n

n ≥ 1.

gn = c (1+±)n’1 (1+β)n’1

(n’1)! (γ)n

So we can compute the corresponding Floquet solution using the perhaps

most famous function we de¬ne in this chapter:

Hypergeometric Function

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

∞

(±)n (β)n n

|z| < 1,

F (±, β; γ; z) = z,

n! (γ)n

n=0

is called hypergeometric function. It is a solution of a scalar

second-order ODE, called hypergeometric di¬erential equation,

introduced in the exercises below.

For arbitrary dimension ν, a special case where one can compute Floquet

solutions in closed form occurs for A, B as in (2.8), under some additional

generic assumptions upon the parameters. For details we again refer to [13];

here we only mention that for these calculations one has to use the following

other well-known special functions:

Generalized Hypergeometric Functions

For m ≥ 1, ±j ∈ C , 1 ¤ j ¤ m, and βj ∈ C \ {0, ’1, ’2, . . .},

1 ¤ j ¤ m ’ 1, the function

∞

(±1 )n · . . . · (±m )n

zn

F (±1 , . . . , ±m ; β1 , . . . , βm’1 ; z) =

n! (β1 )n · . . . · (βm’1 )n

n=0

(with radius of convergence of the series at least equal to 1), is

called generalized hypergeometric function. It arises in solutions

of a scalar mth-order ODE, called generalized hypergeometric

di¬erential equation, introduced in the exercises below.

Exercises: In the following exercises, let ±, β, γ, resp. ±j , βj be as

in the de¬nition of the hypergeometric, resp. generalized hypergeometric

functions, and let δ denote the di¬erential operator z(d/dz).

1. Verify that the hypergeometric ODE

z(1 ’ z) y + (γ ’ (± + β + 1)z) y ’ ±β y = 0

has the solution F (±, β; γ; z), |z| < 1.

2.4 Systems with General Spectrum 27

2. Show that the function F (±1 , . . . , ±m ; β1 , . . . , βm’1 ; z) satis¬es the

following scalar ODE of mth order, called the generalized hypergeo-

metric di¬erential equation:

d

[(δ + ±1 ) · . . . · (δ + ±m ) ’ (δ + β1 ) · . . . · (δ + βm’1 ) ] y = 0.

dz

3. Use Theorem 1 (p. 4) to conclude that F (±1 , . . . , ±m ; β1 , . . . , βm’1 ; z)

can be holomorphically continued into the complex plane with a single

cut from 1 to in¬nity, which usually is made along the positive real

axis.

4. Show

1

(1 ’ t)γ’β’1 tβ’1

“(γ)

F (±, β; γ; z) = dt, (2.10)

“(β) “(γ ’ β) (1 ’ zt)±

0

for z ∈ C \ [1, ∞) and Re γ > Re β > 0.

5. For Re (γ ’ ± ’ β) > 0, show

∞

“(γ) “(γ ’ ± ’ β)

(±)n (β)n

= . (2.11)

“(γ ’ ±) “(γ ’ β)

n! (γ)n

n=0

6. Show F (±, β; γ; z) = (1 ’ z)γ’±’β F (γ ’ ±, γ ’ β; γ; z).

2.4 Systems with General Spectrum

Throughout this section, let a system (2.1) with general spectrum be given.

We shall solve the same three problems stated at the beginning of this

chapter; however, the solutions will be less direct, and some modi¬cations

in the statements and proofs of the results are necessary. Our treatment

is very similar to Gantmacher™s [105], although the proofs are of a slightly

di¬erent ¬‚avor.

The main tools in this section will be so-called analytic transformations,

introduced and studied by Birkho¬ [54]:

A square matrix-valued function T (z) will be called an analytic trans-

formation if it is holomorphic in a neighborhood of the origin, and so that

∞

det T (z) = 0 there. This obviously is equivalent to T (z) = 0 Tn z n for

|z| su¬ciently small, and det T0 = 0. If Tn = 0 for every n ≥ 1, we shall

sometimes speak of a constant transformation.

Given an analytic transformation T (z), we set x = T (z) x. Then x is a

˜

solution of (2.1) if and only if x solves

˜

z x = B(z) x,

˜ ˜ (2.12)

28 2. Singularities of First Kind

with B(z) given by

zT (z) = A(z) T (z) ’ T (z) B(z). (2.13)

Given T (z) and (2.1), we refer to (2.12) as the transformed system. The

matrix B(z) is again holomorphic near the origin, but possibly on a smaller

circle. In particular, the transformed system has a singularity of ¬rst kind

there. For later use, we mention that in case of A(z) having a pole of order

r at the origin, then so does B(z). Therefore, analytic transformations

preserve the Poincar´ rank of systems at the origin. Note that we shall use

e

the term “analytic transformation” both for the matrix T (z) and for the

change of variable x = T (z) x, but we think this will not lead to confusion.

˜

The special form of T (z) implies that solutions of both systems (2.1) and

(2.12) behave the same near the origin, e.g., show the same monodromy

behavior. This is why we call two systems (2.1) and (2.12) analytically

equivalent, if we can ¬nd an analytic transformation T (z) satisfying (2.13).

Note that (2.13), for given A(z) and B(z), is nothing but a linear system

of ODE of Poincar´ rank r = 1 for the entries of T (z). To check analytic

e

equivalence requires ¬nding a solution T (z) that is holomorphic at the

origin, with T (0) invertible.

Given (2.1), we shall see that we can construct an analytic transfor-

mation so that the task of ¬nding solutions of the transformed system

is easier than for the original one. We have already seen this happen on

p. 21, where a transformation x = e»z x helped to ¬nd solutions of the

˜

two-dimensional con¬‚uent hypergeometric system. Theorem 5 (p. 19) may

be restated as saying that systems with good spectrum are analytically

equivalent to (2.12) with B(z) ≡ A0 , which is an elementary system with

fundamental solution z A0 . For general spectrum we shall prove that (2.1)

is analytically equivalent to a reduced system in the sense of Section 1.5,

with diagonal blocks which all have good spectrum. To do so, we show the

following lemma, concerning a nonlinear system of ODE of a very special

form:

Lemma 2 Given a system (2.1), assume that A(z) can be blocked as

A11 (z) A12 (z)

A(z) = ,

A21 (z) A22 (z)

so that A12 (0) = 0, and that the square matrices A11 (0) and A22 (0)+nI, for

every n ∈ N, have disjoint spectra. Then for su¬ciently small ρ > 0 there

˜

exists a unique matrix-valued holomorphic function T12 (z), with T12 (0) = 0

and

A11 (z) T12 (z) ’ T12 (z) A22 (z)

zT12 (z) =

’T12 (z) A21 (z) T12 (z) + A12 (z), |z| < ρ.

˜ (2.14)

2.4 Systems with General Spectrum 29

(jk)

∞ ∞

Proof: Expanding Ajk (z) = 0 An z n , T12 (z) = Tn z n and insert-

1

ing into (2.14), we obtain the recursions

n’1

(22) (11) (11) (22)

+ nI) ’ (An’m Tm ’ Tm An’m )

Tn (A0 A0 Tn =

m=1

n’µ