The Maillet--Malgrange type theorem for generalized power series

0.76 MB·

by  Renat Gontsov

The Maillet–Malgrange type theorem for generalized power
series
R.R.Gontsov, I.V.Goryuchkina
Abstract
ThereisproposedtheMaillet–Malgrangetypetheoremforageneralizedpowerseries(havingcom-
plex power exponents) formally satisfying an algebraic ordinary differential equation. The theorem
describes the growth of the series coefficients.
1 Introduction
6
1
0 Let us consider an ordinary differential equation (ODE)
2
F(z,u,δu,...,δmu)=0 (1)
n
a
of order m with respect to the unknown u, where F(z,u ,u ,...,u ) (cid:54)≡ 0 is a polynomial of m+2
J 0 1 m
variables, δ =z d .
1 TheclassicaldMz aillettheorem[10]assertsthatanyformalpowerseriessolutionϕ=(cid:80)∞ c zn ∈C[[z]]
2 n=0 n
of (1) is a power series of Gevrey order 1/k for some k ∈R>0∪{∞}. This means that the power series

]
A f = (cid:88)∞ cn zn
C Γ(1+n/k)
n=0
.
h
converges in a neighbourhood of zero, where Γ is the Euler gamma-function. In other words,
t
a
m |c |(cid:54)ABn(n!)1/k
n
[
for some A,B >0.

1 FirstexactestimatesfortheGevreyorderofapowerseriesformallysatisfyinganODEwereobtained
v by J.-P.Ramis [13] in the linear case,
8
7 Lu=a (z)δmu+a (z)δm−1u+...+a (z)u=0, a ∈C{z}.
m m−1 0 i
7
5
He has proved that such a power series is of the exact Gevrey order 1/k ∈ {0,1/k ,...,1/k }, where
1 s
0
k < ... < k < ∞ are all of the positive slopes of the Newton polygon N(L) of the operator L. This
. 1 s
1 polygon is defined as the boundary curve of a convex hull of a union (cid:83)m X ,
0 i=0 i
6 X ={(x,y)∈R2 |x(cid:54)i, y (cid:62)ord a (z)}, i=0,1,...,m
i 0 i
1
: (see Fig. 1). The exactness of the Gevrey order means that there is no k(cid:48) >k such that the power series
v
i is of the Gevrey order 1/k(cid:48).
X
r
a
Figure 1: The Newton polygon N(L) with two positive slopes k = (ord a −ord a )/(j −i), k =
1 0 j 0 i 2
(ord a −ord a )/(m−j).
0 m 0 j
1
TheresultofRamishasbeenfurthergeneralizedbyB.MalgrangeandY.Sibuyaforanon-linearODE
of the general form (1).
Theorem 1 (Malgrange [11]). Let ϕ ∈ C[[z]] satisfy the equation (1), that is F(z,Φ) = 0, where
Φ=(ϕ,δϕ,...,δmϕ), and ∂F (z,Φ)(cid:54)=0. Then ϕ is a power series of Gevrey order 1/k, where k is the
∂um
least of all the positive slopes of the Newton polygon N(L ) of a linear operator
ϕ
m
(cid:88) ∂F
L = (z,Φ)δi
ϕ ∂u
i
i=0
(or k =+∞, if N(L ) has no positive slopes).
ϕ
The refinement of Theorem 1 belongs to Y.Sibuya [15, App. 2]. This claims that ϕ is a power series
of the exact Gevrey order 1/k ∈ {0,1/k ,...,1/k }, where k < ... < k < ∞ are all of the positive
1 s 1 s
slopes of the Newton polygon N(L ).
ϕ
In the paper we study generalized power series solutions of (1) of the form
∞
(cid:88)
ϕ= c zsn, c ∈C, s ∈C, (2)
n n n
n=0
with the power exponents satisfying conditions
0(cid:54)Res (cid:54)Res (cid:54)..., lim Res =+∞
0 1 n
n→∞
(the latter, in particular, implies that a set of exponents having a fixed real part is finite).
Note that substituting the series (2) into the equation (1) makes sense, as only a finite number of
terms in ϕ contribute to any term of the form czs in the expansion of F(z,Φ)=F(z,ϕ,δϕ,...,δmϕ) in
powers of z. Indeed, δjϕ=(cid:80)∞n=0cnsjnzsn and an equation s=sn0 +sn1 +...+snl has a finite number
of solutions (s ,s ,...,s ), since 0 (cid:54) Res → +∞. Furthermore, for any integer N an inequality
n0 n1 nl n
Re(s +s +...+s )(cid:54)N has also a finite number of solutions, so that powers of z in the expansion
n0 n1 nl
of F(z,Φ) can be ordered by the increasing of real parts. Thus, one may correctly define the notion of
a formal solution of (1) in the form of a generalized power series. In particular, the Painlev´e III, V, VI
equations are known to have such formal solutions (see [14], [3], [8], [12]).
For the generalized power series (2) one may naturally define the valuation
valϕ=s ,
0
and this is also well defined for any polynomial in z,ϕ,δϕ,...,δmϕ.
The main result of the paper is an analogue of the Maillet theorem (more precisely, of the Malgrange
theorem) for generalized power series.
Theorem 2. Let the generalized power series (2) formally satisfy the equation (1), ∂F (z,Φ) (cid:54)= 0,
∂um
and for each i=0,1,...,m one have
∂F
(z,Φ)=A zλ+B zλi +..., Reλ >Reλ, (3)
∂u i i i
i
where not all A equal zero. Let k be the least of all the positive slopes of the Newton polygon N(L ) (or
i ϕ
k = +∞, if N(L ) has no positive slopes). Then for any sector S of sufficiently small radius with the
ϕ
vertex at the origin and of the opening less than 2π, the series
∞
(cid:88) cn zsn
Γ(1+s /k)
n
n=0
converges uniformly in S.
2
The Newton polygon of L in the case of the generalized power series ϕ is defined similarly to the
ϕ
classical case, as the boundary curve of a convex hull of a union
m
(cid:91)(cid:110) ∂F (cid:111)
(x,y)∈R2 |x(cid:54)i, y (cid:62)Reval (z,Φ) .
∂u
i
i=0
Note that k = +∞ in Theorem 2 if and only if A (cid:54)= 0. In this case ϕ does converge in S, which
m
has been already proved in [6] by the majorant method. Here we consider the case k <+∞ using other
known methods rather than the majorant one (of course, the convergence could also be proved by these
methods).
The first step (Sections 2, 3) consists of representing the generalized power series solution (2) of (1)
by a multivariate Taylor series and follows from the ”grid-basedness” of this formal solution. The latter
means that there are ν1,...,νr ∈ C such that any term czs of ϕ is of the form c(zν1)m1...(zνr)mr =
czm1ν1+...+mrνr, for some m1,...,mr ∈ Z+. An earlier result of this kind is due to Grigoriev, Singer
[7] studying generalized power series solutions with real exponents. According to later results [9], [1],
more general (real) transseries solutions of a polynomial ODE are actually also grid-based (see [5] as
an introduction to transseries). The second step (Section 4) consists of applying the implicit mapping
theorem for Banach spaces of the obtained multivariate Taylor series.
2 Reduction of the ODE to a special form
Aswementionedabove, weproveTheorem2inthecasek <+∞. ThisimpliesA =0intheexpansion
m
(3)of ∂F (z,Φ). Let0(cid:54)p<mbesuchthatA (cid:54)=0andA =0foralli>p. Thentheminimalpositive
∂um p i
slope k of the Newton polygon N(L ) is
ϕ
Reλ −Reλ
k =min i . (4)
i>p i−p
First we will reduce the equation (1) to a special form. This is provided by a transformation
µ
(cid:88)
u= c zsn +zsµv,
n
n=0
withµ(cid:62)0thatwillbechosenlater. Weuseideasof[11]adaptedtothecaseofgeneralizedpowerseries.
The formal solution (2) can be represented in the form
µ
(cid:88)
ϕ= c zsn +zsµψ =ϕ +zsµψ, valψ =s −s .
n µ µ+1 µ
n=0
Taking into consideration the equality δ(zsµψ)=zsµ(δ+sµ)ψ, we have the relations
δi(zsµψ)=zsµ(δ+s )iψ, i=1,...,m.
µ
Therefore, denoting
Φ=(ϕ,δϕ,...,δmϕ)=Φ +zsµΨ, Ψ=(ψ ,ψ ,...,ψ )=(ψ,(δ+s )ψ,...,(δ+s )mψ),
µ 0 1 m µ µ
and applying the Taylor formula to the relation F(z,Φ)=0, we have
m
(cid:88) ∂F
0 = F(z,Φ +zsµΨ)=F(z,Φ )+zsµ (z,Φ )ψ +
µ µ ∂u µ i
i
i=0
1 (cid:88)m ∂2F
+ z2sµ (z,Φ )ψ ψ +.... (5)
2 ∂u ∂u µ i j
i j
i,j=0
3
According to the assumptions of Theorem 2 and definition of the integer p, for each i = 0,1,...,m
the formal series ∂F (z,Φ) is of the form
∂ui
∂F
(z,Φ)=A zλ+B zλi +..., Reλ >Reλ,
∂u i i i
i
where A =0 for all i>p and A (cid:54)=0. Define a polynomial
i p
L(ξ)=A +A (ξ+s )+...+A (ξ+s )p (6)
0 1 µ p µ
of degree p and choose an integer µ(cid:62)0 such that the following three conditions i), ii), iii) hold.
i) L(ξ)(cid:54)=0 ∀ξ with Reξ >0;
ii) Re(s −s )>0;
µ+1 µ
iii) Res >Reλ+2(m−p)k.
µ
Note that for each i=0,1,...,m, the real part of
(cid:18)∂F ∂F (cid:19) (cid:18) (cid:88)m ∂2F (cid:19)
val (z,Φ)− (z,Φ ) =val zsµ (z,Φ )ψ +...
∂u ∂u µ ∂u ∂u µ j
i i i j
j=0
is greater than Res >Reλ. Therefore (see (3) for ∂F (z,Φ)),
µ ∂ui
∂F
∂u (z,Φµ)=Aizλ+B(cid:101)izλ˜i +..., Reλ˜i >Reλ. (7)
i
Moreover, for i>p we have
Reλ˜ (cid:62)Reλ+(i−p)k. (8)
i
Indeed, if B(cid:101)izλ˜i = Bizλi, then the above inequality follows from the definition (4) of the slope k,
otherwise one has Reλ˜ > Res , and (8) follows from the condition iii) above. Thus, (7) and (8) imply
i µ
a decomposition
m
(cid:88) ∂F
z−λ (z,Φ )(δ+s )i =L(δ)+L(cid:48)(z,δ), (9)
∂u µ µ
i
i=0
where the polynomial L is defined by the formula (6), and exponents α in the monomials zα(δ+s )i of
µ
the operator L(cid:48)(z,δ) satisfy the inequalities
Reα>0, Reα(cid:62)(i−p)k.
From the relation (5), condition ii) and inequality Res >Reλ it follows that
µ
RevalF(z,Φ )>Re(s +λ). (10)
µ µ
Finally, dividing the relation (5) by zsµ+λ and using (9), (10), and the condition iii), we obtain the
equality of the form
L(δ)ψ+L(cid:48)(z,δ)ψ+N(z,zνψ ,zνψ ,...,zνψ )=0, ν =(m−p)k, (11)
0 1 m
where the linear differential operators L(δ), L(cid:48)(z,δ) are described above, and N(z,u ,u ,...,u ) is a
0 1 m
finite linear combination of monomials of the form
zβuq0uq1...uqm, β ∈C, Reβ >0, q ∈Z .
0 1 m i +
Thus, the transformation u=ϕµ+zsµv reduces the equation (1) to an equation
L(δ)v+L(cid:48)(z,δ)v+N(z,zνv,zν(δ+s )v,...,zν(δ+s )mv)=0, (12)
µ µ
with a formal solution v =ψ.
Remark 1. The condition i) is not used for the above reduction, but we add it from the beginning
as this will be used in the sequel. In particular, i) implies that the coefficients c , n (cid:62) µ+1, of the
n
formal solution ψ = (cid:80)∞n=µ+1cnzsn−sµ of (12) are uniquely determined by the coefficients c0,...,cµ of
the partial sum ϕ of ϕ (whereas some of c ,...,c themselves may be free parameters).
µ 0 µ
4
3 Representation of a generalized power series solution by a
multivariate Taylor series
Let us define an additive semi-group G generated by a (finite) set consisting of the number ν and all
the power exponents α, β of the variable z containing in the monomials zα(δ+sµ)i, zβuq00uq11...uqmm of
L(cid:48)(z,δ), N(z,u ,u ,...,u ) respectively. Let r ,...,r be generators of this semi-group, that is,
0 1 m 1 l
l
(cid:88)
G={m r +...+m r |m ∈Z , m >0}, Rer >0.
1 1 l l i + i i
i=1
As a consequence of the relation (11) and condition that the real parts of the power exponents α, β are
positive, we have the following auxiliary lemma (details of the proof see in [6, Lemma 2]).
Lemma 1. All the numbers s −s , n(cid:62)µ+1, belong to the additive semi-group G.
n µ
We may assume that the generators r ,...,r of G are linearly independent over Z. This is provided
1 l
by the following lemma [6, Lemma 3].
Lemma 2. There are complex numbers ρ ,...,ρ linearly independent over Z, such that all Reρ >0
1 τ i
and anadditive semi-group G(cid:48) generatedby them containsthe above semi-group G generatedby r ,...,r .
1 l
In what follows, for the simplicity of exposition we assume that G is generated by two numbers:
G={m r +m r |m ∈Z , m +m >0}, Rer >0.
1 1 2 2 1,2 + 1 2 1,2
In the case of an arbitrary number l of generators all constructions are analogous, only multivariate
Taylor series in l rather than in two variables are involved.
We should estimate the growth of the coefficients c of the generalized power series
n
∞
(cid:88)
ψ = c zsn−sµ,
n
n=µ+1
which satisfies the equality (11). According to Lemma 1, all the exponents s −s belong to the semi-
n µ
group G:
s −s =m r +m r , (m ,m )∈M ⊆Z2 \{0},
n µ 1 1 2 2 1 2 +
for some set M such that the map n(cid:55)→(m ,m ) is a bijection from N\{1,...,µ} to M. Hence,
1 2
(cid:88) (cid:88)
ψ = c zm1r1+m2r2 = c zm1r1+m2r2
m1,m2 m1,m2
(m1,m2)∈M (m1,m2)∈Z2+\{0}
(in the last series one puts c =0, if (m ,m )(cid:54)∈M).
m1,m2 1 2
Now we define a natural linear map σ :C[[zG]]→C[[z ,z ]] from the C-algebra of generalized power
1 2 ∗
series with exponents in G to the C-algebra of Taylor series in two variables without a constant term,
(cid:88) (cid:88)
σ : a zγ (cid:55)→ a zm1zm2.
γ γ 1 2
γ=m1r1+m2r2∈G γ=m1r1+m2r2∈G
As follows from the linear independence of the generators r ,r over Z,
1 2
σ(η η )=σ(η )σ(η ) ∀η ,η ∈C[[zG]],
1 2 1 2 1 2
hence σ is an isomorphism. The differentiation δ : C[[zG]] → C[[zG]] naturally induces a linear bijective
map ∆ of C[[z ,z ]] to itself,
1 2 ∗
(cid:88) (cid:88)
∆: a zm1zm2 (cid:55)→ γa zm1zm2,
γ 1 2 γ 1 2
γ∈G γ∈G
5
which clearly satisfies ∆◦σ =σ◦δ, so that the following commutative diagramme holds:
C[[zG]] −→δ C[[zG]]
↓σ ↓σ
C[[z ,z ]] −∆→ C[[z ,z ]]
1 2 ∗ 1 2 ∗
Thus we have the representation
ψ˜=σ(ψ)= (cid:88)c zm1zm2
γ 1 2
γ∈G
of the formal solution ψ of (12) by a multivariate Taylor series, where c = c for every γ =
γ m1,m2
m r +m r . Now we apply the map σ to the both sides of the equality (11),
1 1 2 2
L(δ)ψ+L(cid:48)(z,δ)ψ+N(z,zνψ ,zνψ ,...,zνψ )=0,
0 1 m
and obtain a relation for ψ˜:
L(∆)ψ˜+L(cid:101)(z1,z2,∆)ψ˜+N(cid:101)(z1,z2, z1k1z2k2ψ˜,z1k1z2k2ψ˜1,...,z1k1z2k2ψ˜m)=0, (13)
where ψ˜ =σ(ψ )=(∆+s )iψ˜, and
i i µ
a) L(cid:101)(z1,z2,∆) is a finite linear combination of monomials z1l1z2l2(∆+sµ)i which satisfy
Re(l r +l r )(cid:62)(i−p)k,
1 1 2 2
furthermore L(cid:101)(0,0,∆)≡0;
b) N(cid:101)(z1,z2,u0,...,um) is a polynomial, and N(cid:101)(0,0,u0,...,um)≡0;
c) ν =(m−p)k =k r +k r .
1 1 2 2
4 Banach spaces of multivariate Taylor series and the implicit
mapping theorem
In this section we conclude the proof of Theorem 2 establishing the corresponding required properties
for the multivariate Taylor series ψ˜, which represents the generalized power series ψ and satisfies the
relation (13). We use the dilatation method based on the implicit mapping theorem for Banach spaces.
This was originally used by Malgrange for proving Theorem 1, as well as by C.Zhang [16] in a further
generalization of this theorem for q-difference-differential equations.
Let us define the following Banach spaces Hj of (formal) Taylor series in two variables without a
constant term:
(cid:110) (cid:88) (cid:88) |γ|j (cid:111)
Hj = η = a zm1zm2 | |a |<+∞ , j =0,1,...,p,
γ 1 2 |Γ(γ/k)| γ
γ∈G γ∈G
with the norm
(cid:88) |γ|j
(cid:107)η(cid:107) = |a |.
j |Γ(γ/k)| γ
γ∈G
(ThecompletenessofeachHj ischeckedinawaysimilartothathowonechecksthecompletenessofthe
space l ; see, for example, [4, Ch. 6, §4].) One clearly has Hp ⊂Hp−1...⊂H0 and
2
∆+s :Hj →Hj−1, j =1,...,p.
µ
Therefore, (∆+s )p maps Hp to H0, whereas (∆+s )i for i > p may map Hp outside H0. However,
µ µ
linear operators zl1zl2(∆+s )i with suitable l , l possess the following property.
1 2 µ 1 2
6
Lemma 3. Let l , l be such that Re(l r +l r )(cid:62)(i−p)k. Then
1 2 1 1 2 2
zl1zl2(∆+s )i :Hp →H0
1 2 µ
is a continous linear operator.
Proof. Let η =(cid:80) a zm1zm2 ∈Hp. Then, by definition,
γ∈G γ 1 2
(cid:88) |γ+sµ|p |a |<+∞,
|Γ(γ/k)| γ
γ∈G
therefore
Γ(γ/k)
a = A ,
γ (γ+s )p γ
µ
(cid:80)
where A is an absolutely convergent series. Thus we have
γ∈G γ
(cid:88) (cid:88)
z1l1z2l2(∆+sµ)i(η) = (γ+sµ)iaγz1m1+l1z2m2+l2 = (γ−γ(cid:48)+sµ)iaγ−γ(cid:48)z1m1z2m2 =
γ∈G γ>γ(cid:48)
(cid:88)
= (γ−γ(cid:48)+sµ)i−pΓ(γ/k−γ(cid:48)/k)Aγ−γ(cid:48)z1m1z2m2,
γ>γ(cid:48)
where we write γ >γ(cid:48) for γ =m r +m r , γ(cid:48) =l r +l r , if m (cid:62)l , m (cid:62)l and (m ,m )(cid:54)=(l ,l ).
1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 2
Tochecktheinclusionzl1zl2(∆+s )i(η)∈H0,weshouldprovetheabsoluteconvergenceoftheseries
1 2 µ
(cid:88) Γ(γ/k−γ(cid:48)/k)
(γ−γ(cid:48)+s )i−p A . (14)
µ Γ(γ/k) γ−γ(cid:48)
γ>γ(cid:48)
Since for γ →∞, Reγ >0, and a fixed γ(cid:48), one has (see [2, Ch. I, §1.4])
Γ(γ/k−γ(cid:48)/k)
∼C(γ−γ(cid:48))−γ(cid:48)/k,
Γ(γ/k)
a general term of the series (14) is equivalent to C(γ−γ(cid:48))i−p−γ(cid:48)/kA . This implies the convergence
γ−γ(cid:48)
of this series, as Re(i−p−γ(cid:48)/k)(cid:54)0 under the assumptions of the lemma. We also have
(cid:88)
(cid:107)zl1zl2(∆+s )i(η)(cid:107) (cid:54)C |A |(cid:54)C (cid:107)η(cid:107) ,
1 2 µ 0 1 γ 2 p
γ∈G
whence the continuity of zl1zl2(∆+s )i follows. (cid:3)
1 2 µ
Lemma 4. For any η , η ∈H0, one has η η ∈H0 and (cid:107)η η (cid:107) (cid:54)C(cid:107)η (cid:107) (cid:107)η (cid:107) .
1 2 1 2 1 2 0 1 0 2 0
Proof. Let η =(cid:80) a zm1zm2 and η =(cid:80) b zm1zm2. Then
1 γ∈G γ 1 2 2 γ∈G γ 1 2
(cid:88)(cid:16)(cid:88) (cid:17)
η η = a b zm1zm2,
1 2 γ˜ γ−γ˜ 1 2
γ∈G γ˜<γ
so that
(cid:80)
(cid:107)η η (cid:107) = (cid:88) | γ˜<γaγ˜bγ−γ˜|.
1 2 0 |Γ(γ/k)|
γ∈G
Using the well known relation
Γ(γ˜/k)Γ(γ/k−γ˜/k) (cid:90) 1
= (1−t)γk˜−1tγ−kγ˜−1dt,
Γ(γ/k)
0
we have
(cid:12)(cid:12)(cid:12)Γ(γ˜/k)Γ(γ/k−γ˜/k)(cid:12)(cid:12)(cid:12)(cid:54)(cid:90) 1(1−t)Reγk˜−1tReγ−kγ˜−1dt(cid:54)C
(cid:12) Γ(γ/k) (cid:12)
0
7
for any γ˜ <γ ∈G. Hence,
1 C
(cid:54) ,
|Γ(γ/k)| |Γ(γ˜/k)Γ(γ/k−γ˜/k)|
which implies
(cid:107)η η (cid:107) (cid:54)C (cid:88) (cid:88) |aγ˜| |bγ−γ˜| =C(cid:107)η (cid:107) (cid:107)η (cid:107) .
1 2 0 |Γ(γ˜/k)| |Γ(γ/k−γ˜/k)| 1 0 2 0
γ∈Gγ˜<γ
(cid:3)
Now we conclude the proof of Theorem 2 by the implicit mapping theorem for Banach spaces which
we recall below (see [4, Th. 10.2.1]).
Let E, F, G be Banach spaces, A an open subset of the direct product E × F and h : A → G a
continuously differentiable mapping. Consider a point (x ,y )∈A such that h(x ,y )=0 and ∂h(x ,y )
0 0 0 0 ∂y 0 0
is a bijective linear mapping from F to G.
Then there are a neighbourhood U ⊂E of the point x and a unique continuous mapping g :U →F
0 0 0
such that g(x )=y , (x,g(x))∈A, and h(x,g(x))=0 for any x∈U .
0 0 0
We will apply this theorem to the Banach spaces C, Hp, H0, and to the mapping h:C×Hp →H0
defined by
h:(λ,η)(cid:55)→L(∆)η+L(cid:101)(λz1,λz2,∆)η+N(cid:101)(λz1,λz2, λk1+k2z1k1z2k2η,...,λk1+k2z1k1z2k2(∆+sµ)mη),
with L, L(cid:101), and N(cid:101) coming from (13). This mapping is continuously differentiable by Lemmas 3, 4,
moreover h(0,0)=0 and ∂h(0,0)=L(∆) is a bijective linear mapping from Hp to H0. Therefore, there
∂η
are ρ>0 and η ∈Hp such that
ρ
L(∆)ηρ+L(cid:101)(ρz1,ρz2,∆)ηρ+N(cid:101)(ρz1,ρz2, ρk1+k2z1k1z2k2ηρ,...,ρk1+k2z1k1z2k2(∆+sµ)mηρ)=0.
Making the change of variables (z1,z2)(cid:55)→(zρ1,zρ2), which induces an automorphism η(z1,z2)(cid:55)→η(zρ1,zρ2)
of C[[z ,z ]] commuting with ∆, one can easily see that the above relation implies that the power series
1 2 ∗
ηρ(zρ1,zρ2) satisfies the same equality (13) as ψ˜=(cid:80)γ∈Gcγz1m1z2m2 does. Hence, these two series coincide
(the coefficients of a series satisfying (13) are determined uniquely by this equality) and
(cid:88) cγ zm1zm2
Γ(γ/k) 1 2
γ∈G
has a non-zero radius of convergence. This implies (substitute z1 = zr1, z2 = zr2 remembering that
Rer >0) the convergence of the series
1,2
∞
(cid:88) cγ zγ = (cid:88) cn zsn−sµ
Γ(1+γ/k) Γ(1+(s −s )/k)
n µ
γ∈G n=µ+1
for any z from a sector S of sufficiently small radius with the vertex at the origin and of the opening less
than 2π, whence Theorem 2 follows.
Remark 2. From Theorem 2 one deduces the following estimate for the coefficients c of the formal
n
series solution (2) of (1):
|c |(cid:54)ABResn|Γ(1+s /k)|,
n n
for some A,B >0.
References
[1] M.Aschenbrenner, L.van den Dries, J.van der Hoeven, Asymptotic Differential Algebra and Model
Theory of Transseries, 2015 (arXiv).
8
[2] H.Bateman, A.Erd´elyi, Higher Transcendental Functions, vol. 1, McGraw–Hill, 1953.
[3] A.D.Bruno, I.V.Goryuchkina, Asymptotic expansions of the solutions of the sixth Painlev´e equa-
tion, Trans. Moscow Math. Soc. (2010), 1–104.
[4] J.Dieudonn´e, Foundations of Modern Analysis, Academic Press, 1960.
[5] G.A.Edgar, Transseries for beginners, Real Analysis Exchange 35 (2010), 253–310.
[6] R.R.Gontsov, I.V.Goryuchkina, On the convergence of generalized power series satisfying an alge-
braic ODE, Asympt. Anal. 93:4 (2015), 311–325.
[7] D.Yu.Grigoriev, M.F.Singer, Solving ordinary differential equations in terms of series with real
exponents, Trans. Amer. Math. Soc. 327:1 (1991), 329–351.
[8] D.Guzzetti, Tabulation of Painlev´e 6 transcendents, Nonlinearity 25 (2012), 3235–3276.
[9] J.van der Hoeven, Transseries and Real Differential Algebra, Lect. Notes Math. 1888, Springer,
2006.
[10] E.Maillet, Sur les s´eries divergentes et les ´equations diff´erentielles, Ann. Sci. Ecole Norm. Sup. 3
(1903), 487–518.
[11] B.Malgrange, Sur le th´eor`eme de Maillet, Asympt. Anal. 2 (1989), 1–4.
[12] A.Parusnikova,AsymptoticexpansionsofsolutionstothefifthPainlev´eequationinneighbourhoods
ofsingularandnonsingularpointsoftheequation,In: ”FormalandAnalyticSolutionsofDifferential
and Difference equations”, Banach Center Publ. 97 (2012), 113–124.
[13] J.-P.Ramis, D´evissage Gevrey, Ast´erisque 59/60 (1978), 173–204.
[14] S.Shimomura, A family of solutions of a nonlinear ordinary differential equation and its application
to Painlev´e equations (III), (V) and (VI)., J. Math. Soc. Japan 39 (1987), 649–662.
[15] Y.Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continua-
tion, Transl. Math. Monographs 82, A.M.S., 1990.
[16] C.Zhang, Sur un th´eor`eme du type de Maillet–Malgrange pour les ´equations q-diff´erences-
diff´erentielles, Asympt. Anal. 17:4 (1998), 309–314.
9