$q$-Borel-Laplace summation for $q$-difference equations with two slopes

0.18 MB·English

by  Thomas Dreyfus

q-BOREL-LAPLACE SUMMATION FOR q-DIFFERENCE EQUATIONS
WITH TWO SLOPES.
THOMAS DREYFUSAND ANTONELOY
Abstract. In this paper, we consider linear q-difference systems with coefficients that
are germs of meromorphic functions, with Newton polygon that has two slopes. Then,
we explain how to compute similar meromorphic gauge transformations than those ap-
pearing in thework of Bugeaud, using a q-analogue of the Borel-Laplace summation.
6
1
0
2
Contents

n
u Introduction 1
J 1. Definition of q-analogues of Borel and Laplace transformations. 3

5 2. Local analytic study of q-difference systems. 6
2
3. Main result 8

] References 9
V
C
.
h
t
a
Introduction
m
[ Let q be a complex number with |q| > 1, and let us define σ , the dilatation operator
q
3 that sends y(z) to y(qz). Divergent formal power series may appear as solutions of linear
v q-differencesystemswithcoefficientsinC(z). Thesimplestexampleistheq-Eulerequation
4
9 zσ y+y = z,
q
9
2 that admits as solution the formal power series with complex coefficients
0
1. hˆ := (−1)ℓqℓ(ℓ2+1)zℓ+1.
0 ℓ∈N
X
5
To construct a meromorphic solution of the above equation, we use the Theta function
1
v: Θq(z) := q−ℓ(2ℓ+1)zℓ = (1−q−ℓ−1)(1+q−ℓ−1z)(1+q−ℓz−1),
i
X ℓ∈Z ℓ∈N
X Y
r whichisanalyticonC∗,vanishesonthediscreteq-spiral−qZ := {−qn,n ∈ Z},withsimple
a
zero, and satisfies
σ Θ (z) = zΘ (z) = Θ z−1 .
q q q q
(cid:16) (cid:17)
For λ ∈ C∗ we denote by [λ] the class of λ in C∗/qZ. Then, for all [λ] ∈ C∗/qZ \{[−1]}
the following function hˆ[λ], is solution of the same equation as hˆ and is(cid:16)merom(cid:17)orphic on
Date: June28, 2016.
2010 Mathematics Subject Classification. Primary 39A13.
The first author is funded by the labex CIMI. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
under the Grant Agreement No 648132. The final version of this paper will be published in Journal of
DifferenceEquations and Applications.
1
2 THOMASDREYFUSANDANTONELOY
C∗ with simple poles on the q-spiral −λqZ:
qℓλ 1
hˆ[λ] := × .
1+qℓλ Θ q1+ℓλ
Xℓ∈Z q z
(cid:16) (cid:17)
It is easy to see that it does not depend of the chosen representative λ of the class [λ].
Moregenerally,themeromorphicsolutionsplayamajorroleinGaloistheoryofq-difference
equations,see[Bug12,DVRSZ03,RS07,RS09,RSZ13,Sau00]. IntheusualPicard-Vessiot
theory, the Galois group of a q-difference equation with coefficients in C(z) is defined over
C, but the solutions in the Picard-Vessiot rings involve symbols. On the other hand,
Praagman in [Pra86] has shown the existence of a fundamental solution for such system,
that is an invertible solution matrix, with entries that are meromorphic on C∗. Applying
naively the classical Picard-Vessiot theory with those solutions would give a Galois group
defined on the field of functions invariant under the action of σ , that is, the field of
q
meromorphicfunctionsonthetorusC∗/qZ,ratherthanagroupdefinedonC. Fortunately,
Sauloy has explained how to recover the Galois group with the meromorphic solutions.
Let us consider
(1) σ Y = AY, where A ∈ GL (C({z})),
q m
whichmeansthatAisaninvertible m×mmatrix withcoefficients thataregermsofmero-
morphicfunctionsatz = 0. WearegoingtoassumethattheslopesoftheNewtonpolygon
of (1) belong to Z. See [RSZ13], §2.2, for a precise definition. Then, the work of Birkhoff
and Guenther implies that after an analytic gauge transformation, such system may be
putintoaverysimpleform, thatisintheBirkhoff-Guenthernormalform. Moreover, after
a formal gauge transformation, a system in the Birkhoff-Guenther normal form may be
put into a diagonal bloc system. Then, the authors of [RSZ13] build a set of meromorphic
gauge transformations, that make the same transformation as the formal gauge transfor-
mation, with germs of entries, having poles contained in a finite number of q-spirals of the
form qZα, with α∈ C∗. Moreover, the germs of meromorphic gauge transformations they
build are uniquely determined by the set of poles and their multiplicities.
V. Bugeaud considers the case where the Newton polygon of (1) has two slopes that
are not necessarily entire. Analogously to [RSZ13] she builds meromorphic gauge trans-
formations, but this time, the germs of meromorphic entries have poles on qdZ-spirals, for
some d ∈ N∗, instead of qZ-spirals. See [Bug12].
In the differential case, we also have the existence of formal power series solu-
tions of linear differential equations with coefficients in C({z}), but we may apply
to them a Borel-Laplace summation, in order to obtain meromorphic solutions. See
[Bal94, Ram85, Mal95, vdPS03]. In the q-difference case, although there are several q-
analogues of the Borel and Laplace transformations, see [Abd60, Abd64, Dre15a, Dre15b,
DVZ09, Hah49, MZ00, Ram92, RZ02, Tah14, Zha99, Zha00, Zha01, Zha02, Zha03], we do
not know how to compute in general the meromorphic gauge transformations of [Bug12]
using q-analogues of the Borel and Laplace transformations. Remark that when (1)
has two entire slopes, we are in the framework of [RSZ13], and [Dre15a], Theorem 2.4,
explainshow tocomputethem withq-analogues of theBorel andLaplacetransformations.
See Remark 3.3. Although the q-summation process of [Dre15a] applies for q-difference
equations with two slopes in general, the functions obtained have too many poles to be
optimal. Indeed, if the slopes are 0 and n with n,d coprimes numbers, we expect to find
d
meromorphic gauge transformations having entries with n qdZ spirals of poles of order at
most one, and those of [Dre15a] have n qZ spirals of poles of order at most one. The main
goal of this paper is to compute Bugeaud’s like meromorphic gauge transformations with
q-BOREL-LAPLACE SUMMATION 3
a q-Borel-Laplace summation and with a minimal number of poles.
The paper is organized as follows. In §1 we introduce the q-analogues of the Borel and
Laplace transformations. In §2, we give a brief summary of [Bug12]. The meromorphic
gauge transformations she builds have qdZ-spirals of poles of multiplicity at most n,
for some n ∈ N∗. We prove the existence and the uniqueness of meromorphic gauge
dZ
transformations having q n -spiral of poles of multiplicity at most 1. In §3, we explain
how to compute the latter meromorphic gauge transformations using a q-analogue of the
Borel-Laplace summation.
Acknowledgments. The authors would like the thank the anonymous referee for
permitting them to correct a mistake that was originally made in the first version of the
paper.
1. Definition of q-analogues of Borel and Laplace transformations.
We start with the definition of the q-Borel and the q-Laplace transformations. The
definition of the q-Borel we use comes from [Dre15a]. Note that when µ = 1, we recover
the q-Borel transformation that was originally introduced in [Ram92]. The q-Laplace
we use is slightly different than the transformation defined in [Dre15a]. When q > 1
is real and q is replaced by q1/µ, we recover the q-Laplace transformation and the
functional space introduced in [Zha02], Theorem 1.2.1. Earlier introductions of q-Laplace
transformations can befoundin [Abd60, Abd64]. Thoselatter behave differently than our
q-Laplace transformation, since they involve q-deformations of the exponential, instead of
the Theta function.
Throughout the paper, we will say that two analytic functions f and g are equal, if
there exists U ⊂ C, open connected set, such that f and g are analytic on U and are equal
on U. We fix, once for all, a determination of the logarithm over the Riemann surface of
the logarithm we call log. If α∈ C, and b ∈ C∗, then, we write bα instead of eαlog(b). One
has bα+β = bαbβ, for all α,β ∈ C, and all b ∈ C∗ Let C[[z]] be the ring of formal power
series.
Definition 1.1. Let µ ∈ Q . We define the q-Borel transformation of order µ as follows
>0
Bˆ : C[[z]] −→ C[[ζ]]
µ
−ℓ(ℓ−1)
aℓzℓ 7−→ aℓq 2µ ζℓ.
ℓ∈N ℓ∈N
X X
We are going to make the following abuse of notations. For λ ∈ C∗, and n,d positive
co-prime integers, we denote by [λ] the class of λ in C∗/qdnZ. Let M(C∗,0) be the field
of functions that are meromorphic on some punctured neighborhood of 0 in C∗. More
explicitly, f ∈ M(C∗,0) if and only if there exists V, an open neighborhood of 0, such
that f is analytic on V \{0}.
Definition 1.2. Let µ = nd ∈ Q>0 with n,d positive co-prime integers and [λ] ∈ C∗/qdnZ.
An element f of M(C∗,0) is said to belongs to H[λ] if there exist ε > 0 and a connected
µ
domain Ω ⊂ C, such that:
• z ∈ C∗ z−λqℓ < ε qℓλ ⊂Ω.
ℓ∈[dZ n (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)o
n (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) d
• The function f can be continued to an analytic function on Ω with qn-exponential
growth at infinity, which means that there exist constants L,M > 0, such that for
4 THOMASDREYFUSANDANTONELOY
all z ∈ Ω:
|f(z)| < LΘ (M|z|).
d
|q|n
Note that it does not depend of the chosen representative λ. An element f of M(C∗,0)
is said to belongs to Hµ, if there exists a finite set Σ ⊂ C∗/qdnZ, such that for all
[λ] ∈ C∗/qdnZ \Σ, we have f ∈ H[µλ].
Defin(cid:16)ition 1(cid:17).3. Let µ = nd ∈ Q>0 with n,d positive co-prime integers, [λ] ∈ C∗/qdnZ.
Because of [Dre15a], Lemma 1.3, the following map, which does not depend of the chosen
representative λ, is well defined and is called q-Laplace transformation of order µ:
L[λ] : H[λ] −→ M(C∗,0)
µ µ
f qℓλ
f 7−→ .
ℓX∈dnZ Θqnd (cid:16)qndz+(cid:17)ℓλ
(cid:18) (cid:19)
[λ]
For |z|closeto0,thefunctionL (f)(z)haspolesoforderatmost1thatarecontained
µ
dZ dZ
in the q n -spiral −λq n .
Remark 1.4. The q-Laplace we use is slightly different than the transformation defined in
[Dre15a]. Let µ ∈ Q and q′ := q1/µ. The q′-Laplace transformation of order (1,1) of
>0
[Dre15a] equals to the q-Laplace transformation of order µ of this paper.
The following lemmas, which give basic properties of the Borel and Laplace transfor-
d/n
mations, will be needed in §3. For n,d positive co-prime integers, we set σ := σ .
q qd/n
Lemma 1.5. Let µ := nd ∈Q>0 with n,d positive co-prime integers, and [λ] ∈ C∗/qdnZ.
• Let hˆ ∈ C[[z]]. Then, we have Bˆ zσd/n hˆ = ζBˆ hˆ .
µ q µ
• Let f ∈ H[λ]. Then, we have zσd/n(cid:16)L[λ](f(cid:16))=(cid:17)(cid:17)L[λ](ζf)(cid:16). (cid:17)
µ q µ µ
Proof. Thefirstpointis given by [Dre15a], Lemma1.4. Thesecond pointis a consequence
of [Dre15a], Lemma 1.4, and Remark 1.4. (cid:3)
Remark 1.6. Let hˆ ∈ C[[z]] and f ∈ H[λ]. Iterating n times Lemma 1.5, we find
µ
Bˆµ znσqd hˆ = q−d(n2−1)ζnBˆµ hˆ and znσqdLµ[λ](f)= qd(n2−1)Lµ[λ](ζnf).
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)
Let C{z} be the ring of germs of analytic functions at z = 0.
Lemma 1.7. Let µ := nd ∈ Q>0 with n,d positive co-prime integers, [λ] ∈ C∗/qdnZ. Let
f ∈ C{z}. Then,
• Bˆ (f)∈ H[λ].
µ µ
• L[λ]◦Bˆ (f)= f.
µ µ
Proof. Let us prove briefly the first point. Let us take f = a zℓ ∈ C{z}. It’s easy to
ℓ
ℓ∈N
X
see that its q-Borel transform is defined and analytic on the whole plan C, so we just have
d
to check the qn-exponential growth at infinity. Since f ∈ C{z}, we have the existence of
L,M > 0 such that for all ℓ ∈N, |a |< LMℓ. We have for all z ∈ C,
ℓ
Bˆµ(f)(z) ≤ |aℓ||q|−ℓ(2ℓ−n1)d|z|ℓ
(cid:12) (cid:12) ℓX∈N
(cid:12)(cid:12) (cid:12)(cid:12) ≤ L Mℓ|q|−ℓ(2ℓ−n1)d|z|ℓ
ℓ∈N
X
≤ LΘ (M|z|).
d
|q|n
q-BOREL-LAPLACE SUMMATION 5
Let us now prove the second point. Using the expression of Θ , we find that for all
q
k ∈ Z,
Θ d qkndz = qk(k2−n1)dzkΘ d(z).
qn qn
(cid:16) (cid:17)
Following the definition of the q-Laplace transformation, we obtain that, for all
[λ] ∈ C∗/qdnZ,
L[λ](1) = 1.
µ
We claim that for all a ∈ C, ℓ ∈ N, L[λ] ◦Bˆ azℓ = azℓ. Let us prove the claim by an
µ µ
inductionon ℓ. Thecase ℓ = 0is a straightforw(cid:16)ard(cid:17)consequenceof Bˆ (1) = 1, L[λ](1) = 1,
µ µ
and the C-linearity of the two maps.
Let ℓ ∈ N, and assume that for every a ∈ C, we have L[λ] ◦ Bˆ azℓ = azℓ. Let
µ µ
a ∈ C∗ and let us prove that L[λ] ◦ Bˆ azℓ+1 = azℓ+1. We use L(cid:16)emm(cid:17)a 1.5 to find
µ µ
L[µλ]◦Bˆµ azℓ+1 = azL[µλ]◦Bˆµ q−dℓnzℓ .(cid:16)We us(cid:17)e the induction hypothesis to obtain
(cid:16) (cid:17) (cid:16) (cid:17)
azσqd/nL[µλ]◦Bˆµ q−dℓnzℓ = azσqd/n q−dℓnzℓ = azℓ+1.
(cid:16) (cid:17) (cid:16) (cid:17)
This proves the claim. To prove the lemma, we remark that Bˆ (resp. L[λ]) is an additive
µ µ
morphism between C{z} and H[λ] (resp. H[λ] and M(C∗,0)). (cid:3)
µ µ
We finish this section by giving asymptotic properties of the Borel-Laplace summation.
The definition of q-Gevrey asymptotic is an adaptation of [Dre15a], Definition 1.8, in this
setting.
Definition 1.8. Let µ := nd ∈ Q>0 with n,d positive co-prime integers, [λ] ∈ C∗/qdnZ,
f ∈ M(C∗,0) and fˆ:= fˆzi ∈ C[[z]]. We say that
i
i∈N
X
f ∼[λ] fˆ,
µ
if for all ε,R > 0 sufficiently small, there exist L,M > 0, such that for all k ∈ N and for
all z in
z ∈ C∗ |z| < R \ z ∈ C∗ z+qℓλ < ε qℓλ ,
n (cid:12) o ℓ∈[dZn (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)o
(cid:12) n (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)
we have
k−1
f(z)− fˆizi < LMk|q|k(k2−µ1)|z|k.
(cid:12) (cid:12)
(cid:12) Xi=0 (cid:12)
(cid:12) (cid:12)
For µ ∈ Q>0, we define t(cid:12)(cid:12)he formal q-La(cid:12)(cid:12)place transformation of order µ as follows:
Lˆ : C[[ζ]] −→ C[[z]]
µ
ℓ(ℓ−1)
aℓζℓ 7−→ aℓq 2µ zℓ.
ℓ∈N ℓ∈N
X X
Using Lemma 1.5, for all ℓ ∈ N, and [λ] ∈ C∗/qdnZ, we obtain
L ζℓ = L[λ] ζℓ .
µ µ
(cid:16) (cid:17) (cid:16) (cid:17)
b
6 THOMASDREYFUSANDANTONELOY
Proposition 1.9. Let µ := nd ∈ Q>0 with n,d positive co-prime integers, [λ] ∈ C∗/qdnZ
and f ∈ H[λ]. Then, we have
µ
L[λ](f)∼[λ] L (f).
µ µ µ
Proof. This is a direct consequence of [Dre15a], Proposition 1.9, and Remark 1.4. (cid:3)
b
Corollary 1.10. Let µ := nd ∈ Q>0 with n,d positive co-prime integers, [λ] ∈C∗/qdnZ and
f ∈ C[[z]] with Bˆ (f)∈H[λ]. Then L[λ]◦Bˆ (f)∼[λ] f.
µ µ µ µ µ
2. Local analytic study of q-difference systems.
Let C((z)) be the fraction field of C[[z]]. Let K be a sub-field of C((z)) or M(C∗,0)
stable under σ and let A,B ∈ GL (K). The two q-difference systems, σ Y = AY and
q m q
σ Y = BY are said to be equivalent over K, if there exists P ∈ GL (K), called the gauge
q m
transformation, such that
B = P[A]σq := (σqP)AP−1.
In particular,
σ Y = AY ⇐⇒ σ (PY)= BPY.
q q
Conversely, if there exist A,B,P ∈ GL (K) such that σ Y = AY, σ Z = BZ and
m q q
Z = PY, then
B = P [A] .
σ
q
We remind that C({z}) is the field of germs of meromorphic functions at z = 0. Until
the end of the paper, we fix A∈ GL C({z}) . Let µ := n ∈ Q with n,d positive
m d >0
co-prime integers, and a ∈C∗. We defin(cid:16)e the d(cid:17)times d companion matrix as follows:
0 1
 ... ... 
E := .
n,d,a  ... 1
 
azn 0
 
 
For the proof of the following result, see [Bug12], Theorem 1.18. See also [vdPR07],
Corollary 1.6.
Theorem 2.1. There exist integers n ,...,n ∈ Z, d ,...,d ,m ,...,m ∈ N∗, with
1 k 1 k 1 k
n1 < ··· < nk, gcd(n ,d ) = 1, m d = m, and
d1 dk i i i i
• Hˆ ∈GL C((z)) , P
m
• C ∈ GL (cid:16)(C), (cid:17)
i mi
• a ∈ C∗,
i
such that A = Hˆ Diag C ⊗E , where
i ni,di,ai σ
q
h (cid:16) (cid:17)i
C ⊗E
1 n1,d1,a1
Diag Ci⊗Eni,di,ai :=  ... .
(cid:16) (cid:17) C ⊗E
 k nk,dk,ak
 
The aim of this paper will be to replace this Hˆ by a meromorphic gauge transformation
computed with q-Borel-Laplace transformations.
Assume that k = 2, see Theorem 2.1 for the notation, and let n ,n ,d ,d ,a ,a ,C ,C ,
1 2 1 2 1 2 1 2
given by Theorem 2.1. In [Bug12], V. Bugeaud makes the local analytic classification of
q-difference systems with two slopes. Let us now detail her work. She explains how to
q-BOREL-LAPLACE SUMMATION 7
reducetothecasewhereC ⊗E = I ,thatistheidentity matrixofsize1andwhere
2 n2,d2,a2 1
C is a unipotent Jordan bloc. For r ≥ 1 let us write
1
1 1 0
.. ..
 . . 
U := ∈ GL (C),
r  ... 1 r
 
0 1
 
 
the unipotent Jordan bloc of length r.
Until the end of the paper, we assume that k = 2, C ⊗E = I and
2 n2,d2,a2 1
C = U , for some r ≥ 1.
1 r
Note that dueto theassumption, we have n2 = 0 and n1 < 0. Let n := −n ∈ N∗,d := d ,
d2 d1 1 1
µ := n ∈ Q and a := a . Note that m−1 = d×r. Let us remind that C{z} is the ring
d >0 1
of germs of analytic functions at z = 0. Using [Bug12], Theorem 2.9, we deduce:
Theorem 2.2. There exist
n−1
• W, column vector of size dr with coefficients in Czν,
ν=0
X
• F ∈ GL C({z}) ,
m
such that A = F[B(cid:16)]σ , wh(cid:17)ere:
q
E ⊗U W
B := −n,d,a r .
0 1
(cid:18) (cid:19)
It follows from Theorem 2.1 that we have the existence and the uniqueness of
I Hˆ
dr ∈ GL C[[z]] ,
0 1 m
!
(cid:16) (cid:17)
formal gauge transformation, that satisfies
I Hˆ
B = dr Diag(E ⊗U ,1) .
0 1! −n,d,a r σq
h i
Let us define the finite set Σ ⊂ C∗/qdnZ, as follows
Σ := [λ] ∈ C∗/qdnZ λn ∈ aq(n−21)dqdZ .
(cid:26) (cid:12) (cid:27)
Let M(C∗) be the field of meromorphic func(cid:12)tions on C∗.
(cid:12)
Proposition 2.3. Let [λ] ∈ C∗/qdnZ \Σ. There exists a unique matrix
(cid:16) (cid:17)
I Hˆ[λ]
dr ∈GL M(C∗) ,
m
0 1
!
(cid:16) (cid:17)
I Hˆ[λ]
such that B = dr Diag(E ⊗U ,1) , and for all 1 ≤ i ≤ dr, the mero-
0 1 ! −n,d,a r σq
h i
morphic function of the line number i of the column vector Hˆ[λ], has poles of order at
most 1 contained in the qdnZ-spiral −λq−i+1qdnZ.
Remark 2.4. In [Bug12], Proposition 3.8, a similar result is shown. In her work, the
entry number i of the meromorphic gauge transformation has poles of order at most n
contained in the qdZ-spiral −λq−i+1qdZ. It is worth mentioning that our meromorphic
gauge transformation has the same number of poles, counted with multiplicities, which is
a crucial property.
8 THOMASDREYFUSANDANTONELOY
I Hˆ
Proof. Using B = dr Diag(E ⊗U ,1) , we find,
0 1! −n,d,a r σq
h i
(2) σ Hˆ = (E ⊗U )Hˆ +W.
q −n,d,a r
(cid:16) (cid:17)
Let us write (hˆ ) := Hˆ and (w ) := W. Writing the equations of (2), we
i i=1,...,dr i i=1,...,dr
find for all 1 ≤ i< d, for all 0 ≤ k < r, (we make the convention that hˆ = 0 for i> dr)
i
(3) σ hˆ = hˆ +hˆ +w .
q i+kd i+1+kd i+1+(k+1)d i+kd
And for all 0< k ≤ r,
(4) znσ hˆ = ahˆ +ahˆ +w .
q kd (k−1)d+1 kd+1 kd
For 1 ≤ i ≤ d, let Hˆ := (hˆ ) . With (3) and (4), we find that there exist
i i+kd k=0,...,r−1
W ,...,W ∈(C[z])r such that (2) is equivalent to
1 d
znσdHˆ = aUdHˆ +W , and for 1 < i ≤ d, σ Hˆ = U Hˆ +W .
q 1 r 1 1 q i−1 r i i
Consequently, itissufficienttoprovetheexistenceandtheuniquenessofameromorphic
vector Hˆ1[λ] ∈(M(C∗))r with entries having poles of order at most 1 contained in the qdnZ-
dZ
spiral −λq n , satisfying
znσdHˆ[λ] = aUdHˆ[λ]+W .
q 1 r 1 1
If we put q′ := qd, and z′ := zn, we find that the the case W ∈ C[zn] is an applica-
1
tion of [RSZ13], Proposition 3.3.1, combined with [RSZ13], Theorem 6.1.2. Let us write
W = d−1W zκ with W ∈ C[zn] and consider theuniquemeromorphicsolution Hˆ[λ]
1 κ=0 1,κ 1,κ 1,κ
of znσPqdHˆ1[λ,κ] = aUrdHˆ1[λ,κ] + W1,κ with convenient poles. Then Hˆ1[λ] := dκ−=10Hˆ1[λ,κ]zκ is a
meromorphic solution of znσqdHˆ1[λ] = aUrdHˆ1[λ] +W1 with convenient poPles. Let us prove
the uniqueness. To the contrary, let us assume that there exists another meromorphic
solution with the same properties for the poles. Then the equation znσdY = aUdY has a
q r
dZ
non zero meromorphic solution having poles of order at most 1 contained in the q n -spiral
−λqdnZ. This contradict the uniqueness in the case W1 ∈ C[zn], and proves the result. (cid:3)
3. Main result
Let us keep the notations of the previous section. We now state our main re-
sult. We refer to §1 for the definitions of the Borel and Laplace transformations. Let
[λ] ∈ C∗/qdnZ \Σ. We have seen in the proof of Proposition 2.3, that in order to com-
pute H(cid:16)ˆ[λ] =: (cid:17)hˆ[λ] it is sufficient to compute Hˆ[λ] := hˆ[λ] . Let us
i 1 1+kd
i=1,...,dr k=0,...,r−1
write Hˆ =: hˆ(cid:16) (cid:17) , and Hˆ := hˆ . (cid:16) (cid:17)
i 1 1+kd
i=1,...,dr k=0,...,r−1
(cid:16) (cid:17) (cid:16) (cid:17)
Theorem 3.1. We have Bˆ Hˆ ∈ H[λ] r and
µ 1 µ
(cid:16) (cid:17) (cid:16) (cid:17)
Hˆ[λ] = L[λ]◦Bˆ Hˆ .
1 µ µ 1
(cid:16) (cid:17)
Proof. We remind that we have znσdHˆ[λ] = aUdHˆ[λ]+W for some W ∈ (C[z])r.
q 1 r 1 1 1
We claim that Bˆ Hˆ ∈ H[λ] r. We use Remark 1.6, to find that
µ 1 µ
(cid:16) (cid:17) (cid:16) (cid:17)
(5) q−d(n2−1)ζnBˆµ Hˆ1 = aUrdBˆµ Hˆ1 +Bˆµ(W1).
(cid:16) (cid:17) (cid:16) (cid:17)
q-BOREL-LAPLACE SUMMATION 9
Using the definition of Σ, we deduce that for all κ∈ Z,
λnq−d(n2−1)qκd 6= a.
This proves our claim.
With Lemma 1.7, we obtain that L[λ] ◦ Bˆ (W ) = W . Using additionally (5) and
µ µ 1 1
Remark 1.6, we find that
znσd L[λ]◦Bˆ Hˆ = aUdL[λ]◦Bˆ Hˆ +W .
q µ µ 1 r µ µ 1 1
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)
Moreover, the entries of L[λ] ◦ Bˆ Hˆ have poles of order at most 1 contained in the
µ µ 1
qdnZ-spiral −λqdnZ. But we have seen(cid:16) in(cid:17)the proof of Proposition 2.3, that this implies
Hˆ[λ] = L[λ]◦Bˆ Hˆ .
1 µ µ 1
(cid:16) (cid:17) (cid:3)
Example 3.2. Assume that n = 1, d = 2, r = 1, a = 1, and W = (0,1)t. Then, the first
entry hˆ of Hˆ is solution of
zσ2 hˆ = hˆ +z.
q
We have (cid:16) (cid:17)
ζBˆ hˆ = Bˆ hˆ +ζ,
1/2 1/2
which means that Bˆ hˆ = ζ . Th(cid:16)er(cid:17)efore, for(cid:16)al(cid:17)l [λ] ∈ C∗/q2Z \{[1]} , we find that
1/2 ζ−1
Hˆ[λ] = (hˆ[λ],σ hˆ[λ])t, wh(cid:16)er(cid:17)e hˆ[λ] (cid:16) (cid:17)
q
qℓλ 1
hˆ[λ] = × .
qℓλ−1 Θ q2+ℓλ
ℓX∈2Z q2 z
(cid:16) (cid:17)
Remark 3.3. In [Dre15a], Theorem 2.5, the author defines a meromorphic solution with
another q-analogues of the Borel and Laplace transformation. When d = 1, the two sums
obtained are the same. On the other hand when d 6= 1 the solutions defined here have less
poles than the solutions of [Dre15a]. More precisely, see Theorem 2.1 for the notations,
assume that k = 2 and n = 0. Let d ,n be given by Theorem 2.1. In [Dre15a], it is
2 1 1
shown how to build, with another q-Borel and q-Laplace transformations, meromorphic
Z
gauge transformations similar to §2 but with germs of entries having q|n1|-spirals of poles
d1Z
of order 1. The solutions we compute here have q|n1|-spirals of poles of order 1.
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UnivLyon, UniversitéClaudeBernardLyon 1, CNRSUMR5208, InstitutCamilleJordan,
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