where the coe¬cient matrix is exactly as in (3.1), i.e., is holomorphic for

|z| > ρ and has at worst a pole at in¬nity. The ¬rst, trivial although

important, observation is that the order of the pole of A(z) does not play

the same crucial role as for systems of ODE: If we set x(z) = “(z) x(z), then

˜

the functional equation of the Gamma function implies that x(z) satis¬es

˜

’1

a di¬erence equation exactly like before, but with z A(z) as its coe¬cient

matrix. Thus we learn by repeated application of this argument that we

might restrict ourselves to systems where A(z) is holomorphic at in¬nity,

and A(∞) = 0, but for what we shall have to say, this will be not essential.

However, to avoid degeneracies, we shall assume that det A(z) ≡ 0, so

that x(z + 1) = A(z) x(z) can be solved for x(z), obtaining an equivalent

di¬erence equation in the “backward direction.”

Under the assumptions made above, the formal theory of such a di¬erence

equation is well established [80, 81, 92, 133, 134, 223, 269]: Every such

system has a formal fundamental solution of the form

X(z) = F (z) z Λz eQ(z) z L ,

ˆ ˆ

with:

ˆ

• a formal q-meromorphic transformation F (z), for some q ∈ N,

• a diagonal matrix Λ of rational numbers with denominators equal to

the same number q,

• a diagonal matrix Q(z) of polynomials in z 1/q without constant term

and of degree strictly less than q, and

• a constant matrix L in Jordan canonical form, commuting with Q(z)

and Λ.

Hence, the main di¬erence between formal solutions of di¬erence and dif-

ferential equations is the occurrence of the term z Λz . Indeed, if Λ vanishes

or is a multiple of the identity matrix, then Braaksma and Faber [71] have

ˆ

shown that F (z) is (k1 , . . . , kp )-summable in all but countably many direc-

tions, with levels 1 = k1 > . . . > kp > 0 that are determined by Q(z)

in exactly the same fashion as for ODE. If Λ contains several distinct en-

tries on the diagonal, however, things get considerably more complicated

because of the occurrence of a new level, commonly denoted as level 1+ .

A simple but very instructive example showing this phenomenon may be

13.3 Singular Perturbations 201

found in Faber™s thesis [102]. Roughly speaking, this example shows that in

presence of level 1+ one can no longer restrict to using Laplace transform

integrating along straight lines, but has to allow other paths of integration.

For a general presentation, see Chapter 3 of the said thesis, containing joint

work of Braaksma, Faber, and Immink.

13.3 Singular Perturbations

Let us consider a system of ODE of the form

µσ x = g(z, x, µ),

where g(z, x, µ) is as in Section 13.1, but additionally depends on a parame-

ter µ, and σ is a natural number. Analysis of the dependence of solutions on

µ is referred to as a singular perturbation problem. Under suitable assump-

tions on the right-hand side, such a system will have a formal solution

∞

x(z, µ) = n=0 xn (z) µn , with coe¬cients xn (z) given by di¬erential re-

ˆ

cursion relations. In general, the series is divergent, and classically one has

tried to show existence of solutions of the above system that are asymptotic

to x(z, µ) when µ ’ 0 in some sectorial region. Very recently, one has begun

ˆ

to investigate Gevrey properties of x, or discuss its (multi-)summability.

ˆ

Of the recent articles containing results in this direction, we mention Wal-

let [278, 279], Canalis-Durant [76], and Canalis-Durant, Ramis, Sch¨fke, a

and Sibuya [77].

Since here we meet power series whose coe¬cients are functions of an-

other variable, the situation is very much analogous to that of formal so-

lutions of partial di¬erential equations, which we are investigating in the

following section. Here, we shall brie¬‚y look at a very simple example,

already discussed by Ecalle:

The inhomogeneous equation µ x = x ’ f (z) has the formal solution

∞

x(z, µ) = 0 µn f (n) (z), for arbitrary f (z) which we assume holomorphic

ˆ

near the origin, say, for |z| < ρ. According to Cauchy™s formula, the coe¬-

cients f (n) (z) grow roughly like n!, so it is natural to study 1-summability

of this series. The formal Borel transformation of order k = 1 of this series

equals the power series expansion of f (z + µ) about the point z. Conse-

ˆˆ

quently, holomorphic continuation of B1 x is equivalent to that of f . Hence,

the series x is 1-summable in a direction d if and only if the function f

ˆ

admits holomorphic continuation into a sector S(d, δ), for some δ > 0, and

is of exponential growth at most 1 there. More generally, one can prove a

result on the multisummability of x that is completely analogous to Theo-

ˆ

rem 63 (below) for the heat equation. In any case, multisummability of x ˆ

is always linked to explicit conditions on the function f (z). This indicates

that for power series of several variables, the notion of multisummability

that has been developed in this book is not general enough. This is due

202 13. Applications in Other Areas, and Computer Algebra

to the fact that here we can only treat one variable at a time, while the

remaining ones are treated as parameters.

13.4 Partial Di¬erential Equations

There is a classical theory discussing convergence of power series solutions

for Cauchy problems for certain classes of partial di¬erential equations

(PDE). Recently, e¬orts have been made to show that formal power series

arising as solutions of such problems have a certain Gevrey order; for such

results, see [108, 190“194, 213] and the literature cited there. There are

also results, e.g., by Ouchy [214], showing that such formal solutions are

asymptotic representations of proper solutions of the underlying equation.

Not much is known so far about summability of formal solutions of such

problems: Lutz, Miyake, and Sch¨fke [176] obtained a ¬rst result for the

a

complex heat equation, which has been generalized in [27]. Very recent work

by Balser and Miyake [42] treats an even more general case, showing that

the results obtained are not so much dependent on having a formal solution

of a partial di¬erential equation, but carry over to series whose coe¬cients

are given by certain di¬erential recursions.

Here, we brie¬‚y investigate the following situation: Given a function •(z),

analytic in some G containing the origin, try to ¬nd another function u(t, z)

of two complex variables, in some subset of C 2 near the origin, such that

‚2

‚

u = 2 u, u(0, z) = •(z). (13.2)

‚t ‚z

∞

Obviously, (13.2) has the formal solution u(t, z) = n=0 tn •(2n) (z)/n! In

ˆ

an arbitrary compact subset K ‚ G, one can show, using Cauchy™s formula

for derivatives, that •(2n) K = supz∈K |•(2n) (z)| ¤ cn “(1 + 2n), for su¬-

ciently large c. Hence u can be regarded as a formal power series in t with

ˆ

coe¬cients in the Banach space E K of functions that are bounded on K

and analytic in its interior. The above estimate implies u ∈ E K [[t]]1 . Thus,

ˆ

it is natural to ask for 1-summability of u. If • happens to be entire, how-

ˆ

ever, the above estimate upon its derivatives may be improved, and then u ˆ

may be summable of another type. Indeed, one can show that for arbitrary

k1 > . . . > kq > 1/2, under certain explicit conditions, best expressed in

the coe¬cients of the power series expansion •(z) = •n z n , one even has

(k1 , . . . , kq )-summability of x. In detail, the following holds [27]:

ˆ

∞

•n z n for |z| < ρ, and de¬ne u as above,

Theorem 63 Let •(z) = ˆ

0

resp. for 0 ¤ j ¤ 2:

∞ ∞

(2n + j)! n ˆ±

ˆ •n “(1 + n/2) (±t)n .

ψj (t) = •2n+j t , ψ (t) =

n!

0 0

13.4 Partial Di¬erential Equations 203

Given any type of multisummability k = (k1 , . . . , kν ) with kν ≥ 1, and

any multidirection d = (d1 , . . . , dν ), admissible with respect to k, let 2k =

(2k1 , . . . , 2kν ), d/2 = (d1 /2, . . . , dν /2). Then the following statements are

equivalent:

(a) For every K ‚ D(0, ρ), u ∈ E K {t}k,d .

ˆ

ˆ

(b) For 0 ¤ j ¤ 1, ψj ∈ C {t}k,d .

(c) ψ ± ∈ C {t}2k,d/2 .

ˆ

Proof: Assume (a). For K1 ‚ D(0, ρ), let K2 ‚ D(0, ρ) be so that K1

is contained in the interior of K2 . Then Cauchy™s formula for the deriva-

tive implies that D = d/dz is a bounded linear operator from E K2 into

E K1 . From Theorem 52 (p. 171) we conclude that u ∈ E K2 {t}k,d im-

ˆ

∞ (2n+1)

(z)/n! ∈ E K1 {t}k,d . Taking z = 0 and us-

plies uz (t, z) =

ˆ 0•

ing Exercise 4 on p. 172, this implies (b). From Theorem 51 (p. 166)

we obtain that (b) is equivalent to ψj (t2 ) being in C {t}2k,d/2 . Apply-

ˆ

ing Theorem 50 (p. 164) and Exercise 4 on p. 109, one can then show

∞ ∞

equivalence with 0 “(1 + n) •2n t2n and 0 “(3/2 + n) •2n+1 t2n be-

ing in C {t}2k,d/2 . This in turn is equivalent to (c). This leaves to con-

clude (a) from either (b) or (c): To do so, ¬rst observe that, as a con-

sequence of Theorem 50 (p. 164), it su¬ces to consider the case ν = 1.

∞ ˆ ˆ ˆ

In this situation, verify that u(t, z) = m=0 z m ψm (t), with ψ0 (t), ψ1 (t)

ˆ

ˆ(µ)

ˆ

as in (b), and ψm (t) = ψj (t)/m! , for m = 2µ + j, 0 ¤ j ¤ 1. So

(µ)

ˆ ˆ

ψm ∈ C {t}k,d , and ψm (t) = (Sk,d ψm )(t) = ψj (t)/m! in some sectorial

region G = G(d, ±), ± > π/k, independent of m. From Proposition 9 (p. 70)

¯

we obtain for every closed subsector S ‚ G existence of constants C, c > 0

¯

with |ψm (t)| ¤ C c “(1 + m s)/m!, t ∈ S, m ≥ 0, for s = (1 + 1/k)/2.

m

∞

Owing to k ≥ 1, the series u(t, z) = m=0 z m ψm (t) converges uniformly

(in two variables) for |z| ¤ ρ, ρ > 0 su¬ciently small and t ∈ S 1 . So

¯

u(t, z) ∈ H(G, E K ), for every K as in (a). Di¬erentiating with respect to

t, one can show

∞ (µ+n) (µ+n)

2µ ψ0 (t) 2µ+1 ψ1 (t)

’ •(2n) (z),

n

‚t u(t, z) = z +z

(2µ)! (2µ + 1)!

µ=0

as t ’ 0. Hence we ¬nd that u(t, z) has u(t, z) as its asymptotic expansion.

ˆ

To show that this expansion is of Gevrey order k, we use Proposition 9

(p. 70) again to obtain

∞

|‚t u(t, z)| ¤ C (c |z|)m “(1 + s(m + n))/m!

n

m=0

1 In fact, if k > 1, i.e., •(z) entire, convergence takes place for every z.

204 13. Applications in Other Areas, and Computer Algebra

∞

xsn exp[’x + c|z| xs ] dx.

=C

0

The integral can be bounded by “(1+sn) times some constant to the power

2

n, completing the proof.

It is worthwhile observing that for t = 0 the series u(t, z) in fact converges

for every z, and hence represents an entire function in z that may be seen

to be of exponential size at most 2. This coincides with the classical result

ˆ

on the convergence of u(t, z) (see below). The formal series ψj (z) resp.

ˆ

ψ ± (z) are explicitly related to •(z), and using acceleration and Laplace

ˆ

operators, one can explicitly reformulate (a), (b) in terms of transforms

of the function •(z). However, it is in general di¬cult to check (a) or (b)

directly through investigation of •(z), except for the special case of ν = 1,

k = k1 = 1. In this situation, the above theorem essentially coincides with

the result obtained by Lutz, Miyake, and Sch¨fke [176]: Conditions (a) or

a

(b) then are equivalent to •(z) admitting holomorphic continuation into

small sectors bisected by rays arg z = d and arg z = d + π, and being of

exponential size not more than 2 there.

As follows from the proof of the above theorem, in case of kν > 1 we have

summability of u(t, z) for every z. For kν = 1, however, summability takes

ˆ

place only in a disc whose radius in general is smaller than the radius of

convergence of •(z). Since k-summability in all directions is equivalent to

ˆ

convergence, we obtain as a corollary of the above theorem that convergence

of u(t, z) is equivalent to the initial condition being an entire function of

ˆ

exponential size at most two. This, however, is a well-known classical result.