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general setting: We shall consider a system (3.1) (p. 37), where A(z) is a
rational matrix function with poles only at the origin and in¬nity. Moreover,
we shall make the following additional assumptions upon the nature of the
singularities at the origin resp. in¬nity:

1. The origin is supposed to be a regular-singular point of the system,
but may not be a singularity of ¬rst kind. Moreover, assume that a
194 12. Other Related Questions


fundamental solution of the form X(z) = S(z) z M , S(z) = 0 Sn z n ,
has been computed. Note that, owing to the absence of other ¬nite
singular points, the power series automatically has an in¬nite radius
of convergence; hence S(z) is an entire function, and det S(z) = 0 for
every z = 0.

2. In¬nity is supposed to be an essentially irregular singularity with an
ˆ
HLFFS (F (z), Y (z)), satisying the assumptions in Section 9.3. Note
that these restrictions are without loss of generality, since some easy
normalizing transformations can be used to make them hold. Also,
recall from Section 9.3 the de¬nition of the associated functions and
their behavior in the cut plane C d , for every nonsingular direction d.

Under these assumptions, let Xj (z) = Fj (z) Y (z) be the normal solutions
of highest level. Then there exist unique invertible matrices „¦j , so that
X(z) = Xj (z) „¦j , j ∈ Z. What we are going to show is how the central
connection matrices „¦j can be computed via an analysis of some functions
Ψ(u; s; k), corresopnding to the HLNS via Laplace transform.
Let k ∈ Z, and recall the de¬nition of j — (k) from p. 147. For ± with
dj — (k)’1 ’ π/(2r) < ± < dj — (k) + π/(2r), consider the integral
∞(±)
r r
z s’1 X(z) ez u
Ψ(u; s; k) = dz.
2πi 0

The assumptions made above imply that X(z) is of moderate growth at the
origin, and of exponential growth at most r in arbitrary sectors at in¬nity.
Therefore, the integral converges absolutely and locally uniformly for Re s
su¬ciently large and u in a sectorial region (near in¬nity) of opening π
and bisecting direction π ’ r ±. We have X(z) = S(z) z M , with an entire
function S(z) of exponential growth at most r, so that it is justi¬ed to
termwise integrate the power series expansion of S(z). Making a change of
variable z r u = eiπ x in the above integral, we obtain for s and u as above

1
Sn “([(n + s) I + M ]/r) (eiπ /u)[(n+s) I+M ]/r , (12.2)
Ψ(u; s; k) =
2πi n=0

where the matrix Gamma function “(A) here is de¬ned by the integral

xA’I e’x dx.
“(A) =
0

Observe that this integral converges absolutely for every matrix A whose
eigenvalues all have positive real parts. Integrating by parts, we can show
A “(A) = “(I +A). Using this, it is possible to extend the de¬nition of “(A)
to matrices having no eigenvalue equal to a nonpositive integer. Therefore,
the expansion (12.2) can serve as holomorphic continuation of Ψ(u; s; k),
12.5 Central Connection Problems 195

with respect to s, to become a meromorphic function of s with poles at the
points
s + µ = ’j, j ∈ N0 .
This will, however, not be needed here.
For the auxiliary functions, de¬ned on p. 147, we may choose z0 = 0,
whenever Re s is large enough. Doing so, we ¬nd
µ
¦— (u; s; k) „¦(k) ,
Ψ(u; s; k) = m m
m=1

where „¦m (k) denotes the mth row of blocks of „¦j — (k) , when this matrix is
blocked of type (s1 , . . . , sµ ). This shows that Ψ(u; s; k) is holomorphic in
C d . For its singular behavior at the points un , recall that ¦— (u; s; k) =
m
hol(u ’ un ) for m = n, while we have shown in the proof of Theorem 46
— ’2πi (sI+Ln )
r’1
=0 ¦n (u; s; k + ) = ¦n (u; s; k) (I ’ e ) + hol(u ’
(p. 149) that
un ). This shows:
Theorem 62 Under the assumptions made above, we have for every s with
Re s large and so that (I ’ e2πi (sI+Ln ) )’1 exists,
r’1
¦n (u; s; k + ) (I ’ e’2πi (sI+Ln ) )’1 „¦(k) + hol(u ’ un ),
Ψm (u; s; k) = n
=0

for every n = 1, . . . , µ.
This identity shows that the central connenction problem can theoreti-
cally be solved as follows: Compute the matrix Ψ(u; s; k), either by its in-
tegral representation or the convergent power series (12.2), for u and Re s
su¬ciently large. Then, continue the function with respect to u to the sin-
(k)
gularities un , and there use the above identity to compute „¦n . Doing this
for every n allows to compute the matrix „¦j — (k) , linking the fundamental
solution X(z) to the corresponding normal solution of highest level.
Without going into detail, we mention that given two matrices „¦j — (k) and
„¦j — (k+1) , one can compute the Stokes multipliers Vj , for j — (k + 1) + 1 ¤
j ¤ j — (k), by factoring „¦j — (k+1) „¦’1 as in the exercises in Section 9.2.
j — (k)
Consequently, if we have computed „¦j — (k+ ) , for = 0, . . . , r ’ 1, then all
Stokes™ multipliers of highest level can be found explicitly. However, the
knowledge of the Stokes multipliers is not su¬cient to ¬nd the central con-
nection matrices: Assume that we had chosen X(z) so that exp[2πi M ] were
in Jordan normal form. Moreover, assume all Stokes™ multipliers of highest
level are known. Then the monodromy factor exp[2πi Mj ] for Xj (z) is given
by (9.5). So by continuation of the relation X(z) = Xj (z) „¦j about in¬n-
ity we obtain „¦j exp[2πi M ] = exp[2πi Mj ] „¦j . This then shows that „¦j
is some matrix which transforms exp[2πi Mj ] into Jordan form. However,
such a matrix is not uniquely determined. In the generic situation where
196 12. Other Related Questions

exp[2πi Mj ] is diagonalizable, „¦j is determined up to a right-hand diagonal
matrix factor, and this is exactly the degree of freedom we have in choosing
a fundamental solution X(z) consisting of Floquet solutions. Therefore, the
knowledge of the Stokes multipliers alone does not determine „¦j .
13
Applications in Other Areas, and
Computer Algebra




In this chapter we shall brie¬‚y describe applications of the theory of multi-
summability to formal power series solutions of equations other than linear
ODE. The e¬orts to explore such applications are far from being complete
and shall provide an excellent chance for future research. In a ¬nal section
we then mention recent results on ¬nding formal solutions for linear ODE
with help of computer algebra.
Suppose that we are given some class of functional equations having
solutions that are analytic functions in one or several variables. Roughly
speaking, we shall then address the following two questions:

• Does such a functional equation admit formal solutions that, aside
from elementary functions such as exponentials, logarithms, general
powers, etc., involve formal power series in one or several variables?
Can one, perhaps, even ¬nd a family of formal solutions that is com-
plete in some sense?
• Given such a formal solution, are the formal power series occurring,
if not already convergent, summable in some sense or another, and
if so, are the functions obtained by replacing the formal series with
their sums then solutions of the same functional equation?

Two general comments should be made beforehand: First, it is not at all
clear whether one should in all cases consider formal solutions involving
formal power series “ e.g., one could instead consider formal factorial se-
ries. One one hand, each formal power series can be formally rewritten
as a formal factorial series and vice versa, so both approaches may seem
198 13. Applications in Other Areas, and Computer Algebra

equivalent. On the other hand, however, it may be easier to ¬nd the coe¬-
cients for a factorial series solution directly from the underlying functional
equation, and the question of summation for a factorial series may have an
easier answer than for the corresponding power series. Since we have only
discussed summation of formal power series, we shall here restrict to that
case, but mention a paper by Barkatou and Duval [48], concerning sum-
mation of formal factorial series. The second comment we wish to make
concerns the question of summation of formal power series in several vari-
ables: In the situations we are going to discuss we shall always treat all
but one variable as (temporarily ¬xed) parameters, thus leaving us with a
series in the remaining variable, with coe¬cients in some Banach space. It
is for this reason that we have developed the theory of multisummability in
such a relatively general setting. While this approach is successful in some
situations, there are other cases indicating that one should better look for
a summation method that treats all variables simultaneously, but so far
nobody has found such a method!


13.1 Nonlinear Systems of ODE
Throughout this section, we shall be concerned with nonlinear systems of
the following form:
z ’r+1 x = g(z, x), (13.1)
where r, the Poincar´ rank, is a nonnegative integer, x = (x1 , . . . , xν )T ,
e
ν ≥ 1, and g(z, x) = (g1 (z, x), . . . , gν (z, x))T is a vector of power series
in x1 , . . . , xn . Let p = (p1 , . . . , pν )T be a multi-index, i.e., all pj are non-
negative integers, and de¬ne xp = xp1 · . . . · xpν . Then such a power se-
1 ν

ries can be written as gj (z, x) = gj (z, x1 , . . . , xn ) = p≥0 gj,p (z) xp . The
coe¬cients gj,p (z) are assumed to be given by power series in z ’1 , say,

gj,p (z) = m=0 gj,p,m z ’m . Throughout, it will be assumed that all these
series converge for |z| > ρ, with some ρ ≥ 0 independent of p, while the
series for gj (z, x), for every such z, converges in the ball x < ρ.
As is common for multi-indices, we write |p| = p1 + . . . + pν . If it so
happens that gj,p (z) ≡ 0 whenever |p| ≥ 2, for every j, then (13.1) obviously
becomes an inhomogeneous linear system of ODE, whose corresponding
homogeneous system is as in (3.1) (p. 37). If gj,0 (z) ≡ 0 for every j, then
(13.1) obviously has the solution x(z) ≡ 0, and we then say that (13.1) is
a homogeneous nonlinear system.
It is shown in [24] that homogeneous nonlinear systems have a formal
solution x(z), sharing many of the properties of FFS in the linear case:
ˆ

1. The formal solution x(z) is a formal power series x(z) = x(z, c) =
ˆ ˆ ˆ
p
|p|≥0 xp (z) c in parameters c1 , . . . , cν . Its coe¬cients xp (z) are ¬-
ˆ ˆ
ˆ
nite sums of expressions of the form f (z) z » logk z exp[p(z)], with a
13.2 Di¬erence Equations 199

formal power series f in z ’1 , a complex constant », a nonnegative
ˆ
integer k, and a polynomial p in some root of z. In case of a linear
system, the coe¬cients xp (z) are zero for |p| ≥ 2, so that then x(z, c)
ˆ ˆ
is a linear function of c1 , . . . , cν and corresponds to a FFS.

2. The proof of existence of x(z) follows very much the same steps as
ˆ
in the linear case: One introduces nonlinear analytic, resp. meromor-
phic, transformations and shows that by means of ¬nitely many such
transformations one can, step by step, simplify the system in some
clear sense so that in the end one can “solve” it explicitly. For details,
see [22, 24].

3. The formal power series occurring in the coe¬cients xp (z) all are
ˆ
multisummable, as is shown in [26].

Despite all the analogies to the linear situation, there are two new di¬-
culties for nonlinear systems: For one, it is not clear whether the formal
solution x(z, c) is complete in the sense that every other formal expression
ˆ
solving (13.1) can be obtained from x(z, c) by a suitable choice of the pa-
ˆ
rameter vector c. Moreover, suppose that all the formal power series in
x(z, c) are replaced by their multisums, so that we obtain a formal power
ˆ
series in the parameter vector c, with coe¬cients which are holomorphic
functions in some sectorial region G at in¬nity. It is a well-known fact,
called the small denominator phenomenon, that in general this series di-
verges. Su¬cient conditions for convergence are known; see, e.g., the papers
of Iwano [139, 140] and the literature quoted there. For an analysis of the
nonlinear Stokes phenomenon under a nonresonance condition, compare
Costin [83]. In general, however, it is still open how this series is to be
interpreted.
A related but simpler problem for nonlinear systems is as follows: Sup-
pose that (13.1) has a solution in the form of a power series in z ’1 , then is
this series multisummable? By now, there are three proofs for the answer
being positive, using quite di¬erent methods: Braaksma [70] investigated
the nonlinear integral equations of the various levels which correspond to
(13.1) via Borel transform. Ramis and Sibuya [230] used cohomological
methods, while in [21] a more direct approach is taken, very much like the
proof of Picard-Lindel¨f ™s existence and uniqueness theorem.
o



13.2 Di¬erence Equations
While multisummability is a very appropriate tool for linear and non-
linear ODE, things are more complicated for di¬erence equations, as we
now shall brie¬‚y explain. For a more complete presentation of the theory
of holomorphic di¬erence equations, see the recent book of van der Put
200 13. Applications in Other Areas, and Computer Algebra

and Singer [224]. In his Ph.D. thesis, Faber [102] considers extensions to
di¬erential-di¬erence equations and more general functional equations that
we do not wish to include here. For simplicity we restrict to linear systems
of di¬erence equations, although much of what we say is known to extend
to the nonlinear situation: Let us consider a system of di¬erence equations
of the form

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