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For more details upon this case we refer to .

Exercises: In the Ô¬Ārst two of the following exercises, consider a Ô¬Āxed
conÔ¬‚uent hypergeometric system with A, B as in (2.7). Also, restrict ¬µ1 ‚ą’¬µ2
so that E holds. For the last one, consider m ‚ąą N and parameters ő±j , ő≤j as
in the deÔ¬Ānition of the generalized conÔ¬‚uent hypergeometric function, and
let őī denote the diÔ¬Äerential 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
conÔ¬‚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) conÔ¬‚uent hypergeometric ODE

z y ‚ą’ (z ‚ą’ ő≤)y ‚ą’ ő±y = 0

for ő≤ = 0, ‚ą’1, ‚ą’2, . . . has the solution F (ő±; ő≤; 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 Ô¬Ānd another second-order ODE, closely related to the one above,
bearing the same name.

4. Show Kummer‚Ä™s transformation

F (ő±; ő≤; z) = ez F (ő≤ ‚ą’ ő±; ő≤; ‚ą’z), z ‚ąą C , ő≤ = 0, ‚ą’1, ‚ą’2, . . .

5. For Re ő≤ > Re ő± > 0, show
1
ő“(ő≤)
ezt tő±‚ą’1 (1 ‚ą’ t)ő≤‚ą’ő±‚ą’1 dt.
F (ő±; ő≤; z) =
ő“(ő±) ő“(ő≤ ‚ą’ ő±) 0

6. Verify that Bessel‚Ä™s equation

z 2 y + zy + (z 2 ‚ą’ ¬µ2 )y = 0

has the solution J¬µ (z).
2.3 Hypergeometric Systems 25

7. Show that F (ő±1 , . . . , ő±m ; ő≤1 , . . . , ő≤m ; z) satisÔ¬Āes the following scalar
ODE of (m + 1)st order, called the conÔ¬‚uent hypergeometric diÔ¬Äer-
ential equation:

d
[(őī + ő±1 ) ¬· . . . ¬· (őī + ő±m ) ‚ą’ (őī + ő≤1 ) ¬· . . . ¬· (őī + ő≤m ) ] y = 0.
dz

2.3 Hypergeometric Systems
In this section we consider so-called hypergeometric systems, which are of
the form
(A ‚ą’ zI) x = B x, A, B ‚ąą C őĹ√—őĹ . (2.9)

As shall become clear in the following chapters, (2.9) is intimately related,
via Laplace transform, to a conÔ¬‚uent hypergeometric system with slightly
diÔ¬Äerent A and B. Another relation is through a process called conÔ¬‚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 [ő»1 , . . . , ő»őĹ ], with
not necessarily distinct values ő»k . The system (2.9) then is meromorphic in
C with Ô¬Ārst-order poles at the numbers ő»1 , . . . , ő»őĹ . If all the ő»k are equal, a
change of variable z = u+ő», reduces the system to one treated in Exercise 1
on p. 9, so we exclude this case here. For ő»1 = . . . = ő»őĹ‚ą’1 = ő»őĹ , we shall
compute Floquet solutions of (2.9) using the hypergeometric function resp.
its generalized versions. For őĹ = 3 and three distinct values ő»k , the system,
under some additional assumptions, should be closely related to Heun‚Ä™s
ODE, and the corresponding conÔ¬‚uent one to one of its various conÔ¬‚uent
forms , but to the author‚Ä™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 ő»1 = 0,
ő»őĹ = 1. This shows that in dimension őĹ = 2 we may assume

0 0 ab
A= , B= .
0 1 cd

Putting the system into the form (2.1), we Ô¬Ānd that the Floquet exponents
are ‚ą’a and 0. For simplicity we restrict to a noninteger value for a, so
that in particular E is satisÔ¬Āed. Using the same notation as in the previous
section, the recursion relations for the coeÔ¬Écients of a Floquet solution
with exponent ¬µ = ‚ą’a are

0 = nfn + b gn , (n + 1 ‚ą’ a)gn+1 = c fn + (n + d ‚ą’ a)gn , n ‚Č• 0,
26 2. Singularities of First Kind

with f0 = 1, g0 = 0. Eliminating fn , we obtain n(n + 1 ‚ą’ a) gn+1 =
(n + ő±)(n + ő≤) gn , n ‚Č• 1, with ő± + ő≤ = d ‚ą’ a, ő±ő≤ = ‚ą’cb and the initial
condition g1 = c/(1 ‚ą’ a). With ő≥ = 1 ‚ą’ a, this implies
Ô£ľ
(ő±)n (ő≤)n
Ô£Ĺ
fn = n! (ő≥)n
n ‚Č• 1.
gn = c (1+ő±)n‚ą’1 (1+ő≤)n‚ą’1 Ô£ĺ
(n‚ą’1)! (ő≥)n

So we can compute the corresponding Floquet solution using the perhaps
most famous function we deÔ¬Āne in this chapter:
Hypergeometric Function
For ő±, ő≤ ‚ąą C , ő≥ ‚ąą C \ {0, ‚ą’1, ‚ą’2, . . .}, the function
‚ąě
(ő±)n (ő≤)n n
|z| < 1,
F (ő±, ő≤; ő≥; z) = z,
n! (ő≥)n
n=0

is called hypergeometric function. It is a solution of a scalar
second-order ODE, called hypergeometric diÔ¬Äerential equation,
introduced in the exercises below.
For arbitrary dimension őĹ, 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 ;
here we only mention that for these calculations one has to use the following
other well-known special functions:
Generalized Hypergeometric Functions
For m ‚Č• 1, ő±j ‚ąą C , 1 ‚Č¤ j ‚Č¤ m, and ő≤j ‚ąą C \ {0, ‚ą’1, ‚ą’2, . . .},
1 ‚Č¤ j ‚Č¤ m ‚ą’ 1, the function
‚ąě
(ő±1 )n ¬· . . . ¬· (ő±m )n
zn
F (ő±1 , . . . , ő±m ; ő≤1 , . . . , ő≤m‚ą’1 ; z) =
n! (ő≤1 )n ¬· . . . ¬· (ő≤m‚ą’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
diÔ¬Äerential equation, introduced in the exercises below.

Exercises: In the following exercises, let ő±, ő≤, ő≥, resp. ő±j , ő≤j be as
in the deÔ¬Ānition of the hypergeometric, resp. generalized hypergeometric
functions, and let őī denote the diÔ¬Äerential operator z(d/dz).
1. Verify that the hypergeometric ODE
z(1 ‚ą’ z) y + (ő≥ ‚ą’ (ő± + ő≤ + 1)z) y ‚ą’ ő±ő≤ y = 0
has the solution F (ő±, ő≤; ő≥; z), |z| < 1.
2.4 Systems with General Spectrum 27

2. Show that the function F (ő±1 , . . . , ő±m ; ő≤1 , . . . , ő≤m‚ą’1 ; z) satisÔ¬Āes the
following scalar ODE of mth order, called the generalized hypergeo-
metric diÔ¬Äerential equation:
d
[(őī + ő±1 ) ¬· . . . ¬· (őī + ő±m ) ‚ą’ (őī + ő≤1 ) ¬· . . . ¬· (őī + ő≤m‚ą’1 ) ] y = 0.
dz

3. Use Theorem 1 (p. 4) to conclude that F (ő±1 , . . . , ő±m ; ő≤1 , . . . , ő≤m‚ą’1 ; z)
can be holomorphically continued into the complex plane with a single
cut from 1 to inÔ¬Ānity, which usually is made along the positive real
axis.
4. Show
1
(1 ‚ą’ t)ő≥‚ą’ő≤‚ą’1 tő≤‚ą’1
ő“(ő≥)
F (ő±, ő≤; ő≥; z) = dt, (2.10)
ő“(ő≤) ő“(ő≥ ‚ą’ ő≤) (1 ‚ą’ zt)ő±
0

for z ‚ąą C \ [1, ‚ąě) and Re ő≥ > Re ő≤ > 0.
5. For Re (ő≥ ‚ą’ ő± ‚ą’ ő≤) > 0, show
‚ąě
ő“(ő≥) ő“(ő≥ ‚ą’ ő± ‚ą’ ő≤)
(ő±)n (ő≤)n
= . (2.11)
ő“(ő≥ ‚ą’ ő±) ő“(ő≥ ‚ą’ ő≤)
n! (ő≥)n
n=0

6. Show F (ő±, ő≤; ő≥; z) = (1 ‚ą’ z)ő≥‚ą’ő±‚ą’ő≤ F (ő≥ ‚ą’ ő±, ő≥ ‚ą’ ő≤; ő≥; 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 modiÔ¬Ācations
in the statements and proofs of the results are necessary. Our treatment
is very similar to Gantmacher‚Ä™s , although the proofs are of a slightly
diÔ¬Äerent Ô¬‚avor.
The main tools in this section will be so-called analytic transformations,
introduced and studied by BirkhoÔ¬Ä :
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
‚ąě
det T (z) = 0 there. This obviously is equivalent to T (z) = 0 Tn z n for
|z| suÔ¬Éciently small, and det T0 = 0. If Tn = 0 for every n ‚Č• 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
ňú
solution of (2.1) if and only if x solves
ňú

z x = B(z) x,
ňú ňú (2.12)
28 2. Singularities of First Kind

with B(z) given by

zT (z) = A(z) T (z) ‚ą’ 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 Ô¬Ārst 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¬ī rank of systems at the origin. Note that we shall use
e
the term ‚Äúanalytic transformation‚ÄĚ 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.
ňú
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 Ô¬Ānd 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¬ī rank r = 1 for the entries of T (z). To check analytic
e
equivalence requires Ô¬Ānding 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 Ô¬Ānding 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 = eő»z x helped to Ô¬Ānd solutions of the
ňú
two-dimensional conÔ¬‚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) ‚Č° 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 ‚ąą N, have disjoint spectra. Then for suÔ¬Éciently small ŌĀ > 0 there
ňú
exists a unique matrix-valued holomorphic function T12 (z), with T12 (0) = 0
and

A11 (z) T12 (z) ‚ą’ T12 (z) A22 (z)
zT12 (z) =
‚ą’T12 (z) A21 (z) T12 (z) + A12 (z), |z| < ŌĀ.
ňú (2.14)
2.4 Systems with General Spectrum 29

(jk)
‚ąě ‚ąě
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‚ą’1
(22) (11) (11) (22)
+ nI) ‚ą’ (An‚ą’m Tm ‚ą’ Tm An‚ą’m )
Tn (A0 A0 Tn =
m=1
n‚ą’¬µ
 << ŌūŚšŻšůýŗˇ ŮÚū. 7(ŤÁ 61 ŮÚū.)ő√ňņ¬ňŇÕ»Ň —ŽŚšůĢýŗˇ >>