Gevrey expansions of hypergeometric integrals I

GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I FRANCISCO-JES´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER Abstract. We study integral representations of the Gevrey series solutions of irregular hypergeometric systems. In this paper we consider the case of the systems associated with a one row matrix, for which the integration domains are one dimensional. We prove that any Gevrey series solution along the singular support of the system is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation. February 6, 2013 1. Introduction Hypergeometric systems also known as GKZ systems were introduced in [GGZ87] and [GZK89] as a far reaching generalisation of the Gauss hypergeometric differential equation. They appear as a special family of D-modules and they have been first studied in the regular case. For example in [GKZ90] the authors consider integral representations of the solutions of hypergeometric systems, at generic points of the space, which they call Euler integrals. In the irregular or confluent case, A. Adolphson considers in [A94] other integral representations of solutions which involve exponentials of polynomial functions and appropriate integration cycles. In this paper we develop new aspects in the irregular case namely the link between Gevrey series solutions and holomorphic solutions in sectors following Adolphson’s approach. We want to materialise a Gevrey series solution as an asymptotic expansion in a sector of such an integral solution. Let us fix some notations: Dstands for the complex Weyl algebra of order n, where n≥0 is an integer. Elements in Dare linear partial differential operators with polynomial coefficients. The polynomial ring C[∂] := C[∂1, . . . , ∂n] is a subring of the Weyl algebra D, where the ∂j’s represent the partial derivatives with respect to the variables in the space Cn. The input data for a GKZ system is a pair (A, β) where βis a vector in Cdand A= (aij) = (a(1),· · · , a(n))∈Zd×nis a matrix of rank dwhose jth column is a(j). The toric ideal IA⊂C[∂] is the ideal generated by the family of binomials ∂u−∂vwhere u, v ∈Nnand Au =Av. The ideal IAis a prime ideal and the Krull dimension of the quotient ring C[∂]/IAequals d. Following [GGZ87, GZK89], the hypergeometric ideal associated with the pair (A, β) is : HA(β) = DIA+D(E1−β1, . . . , Ed−βd) where Ei=Pn j=1 aijxj∂jis the ith Euler operator associated with the ith row of A. The corresponding hypergeometric D–module (or hypergeometric system) is the quotient D–module MA(β) := D HA(β). In [GZK89] and [A94] it is proven that any hypergeometric D-module is holonomic. Moreover, a characterization of the regularity of MA(β) is provided in the series of papers [Ho98], [SST] and [SW08]: The holonomic D-module MA(β) is regular if and only if the toric ideal IAis homogeneous for the standard grading in the polynomial ring C[∂]. In particular the condition to be regular for MA(β) is independent of β. The concept of regularity has been studied first in the case of an First author partially supported by MTM2010-19336 and FEDER, FQM333. 1 arXiv:1212.1410v2 [math.AG] 5 Feb 2013 2 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER ordinary differential equation given by an operator P∈Dfor n= 1. Regularity is characterised according to Fuchs theorem by the nullity of the irregularity number, an invariant combinatorially defined from the Newton polygon of P. In [M74] B. Malgrange proved that the irregularity is the dimension of the space of solutions at the origin of Cwith values in the space C[[x]]/C{x}of formal power series modulo convergent ones. Later J.P. Ramis gave a refined version of this result in [R84] calculating the space of solutions of a given Gevrey index again by using the Newton polygon of the operator. The concept of irregularity in higher dimension is considerably more involved but generalizes the above results. Let us consider the structure sheaf OXof a complex manifold X, the sheaf of rings DXof linear differential operators with holomorphic coefficients on Xand denote Od X|Ythe completion of OXalong a smooth hypersurface Y. In [Me90] Z. Mebkhout introduces for a holonomic DX–module Mits irregularity complex along Y, IrrY(M) = RHomDX(M,Od X|Y/OX|Y), and in [LM99] the Gevrey filtration of this complex is introduced and related to other invariants of the system, the algebraic slopes in the sense of Y. Laurent. In the case of hypergeometric D-modules the irregularity sheaves along coordinate subspaces, and Gevrey series solutions are studied and described in [F10] (see also [FC11a, FC11b]). Beforehand A. Adolphson [A94] gave a formula for the dimension of the space of holomorphic solutions at a generic point of the space and for a generic value of the parameter β, and he also described integral representations of solutions of these confluent hypergeometric systems. In the non-confluent case an analogous dimension formula was previously given in [GZK89]. In [ET12] A. Esterov and K. Takeuchi prove that these generic solution spaces are in fact completely described by integral representations along rapid decay cycles as introduced by M. Hien in [Hi07] and [Hi09]. We want to explore the more hidden link between these integral representations and the Gevrey series solutions described in [F10] and [FC11b, FC11a]. In this paper we treat the case of a matrix with one row A= (a1, . . . , an) with 0 < a1<· · · < ana list of co-prime integers. This is the case where the integration cycles are paths, and as explained below the significant Gevrey series are along the hyperplane xn= 0. We prove that any Gevrey series solution of the system can be obtained as the asymptotic expansion of an integral representation along a well chosen path. A specificity of the one row case is that the rank of the space of Gevrey solutions is independent of βand we can treat all the values of β. For a generic βwe only use the space of rapid decay cycles and for special values of β, namely for β∈Z\(Na1+· · · +Nan) we must add an exceptional path, without the rapid decay property. The method of the proof is to reduce the statement to the case of the matrix A= (a, b), using the fact that the restriction is compatible with being a Gevrey solution as well as with taking integrals on a fixed path. In parallel we know by an argument using a Gevrey version of Cauchy-Kovalevskaya theorem that the dimension of the Gevrey solution space is an−1[FC11b]. In the case A= (a, b) the Gevrey expansion is with respect to the second variable x2. Its coefficients depend on the variable x1and are holomorphic in some sector depending on the chosen paths. The main issue is to choose carefully a number of paths of integration that yield a basis of the space of Gevrey series solutions and share a common sector of convergence for these coefficients. This choice appears to be possible if we restrict the range in the argument of the variable x2around a given direction. Here is a summary of the contents of this paper. In Section 2 general facts are given about Gevrey series solutions following [F10]. In the case of a one row matrix we know that the characteristic variety of the hypergeometric system is the union T∗ XX∪T∗ YXof the zero section and of the conormal to the hypersurface Y: (xn= 0). It is therefore sufficient to consider the irregularity sheaf and the germs of Gevrey series solutions with respect to the hyperplane Yand at a generic point (0,...,0, , 0) of Y. We recall from [FC11b] the description of a basis of the space of Gevrey series GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 3 solutions in terms of Γ-series and the proof that the dimension of this space is always equal to an−1. We also describe its behaviour by the restriction operation which consists in omitting variables among x1, . . . , xn−2. In Section 3 we recall the definition of hypergeometric integrals of exponential type and the fact that they are solutions of the system. It is a quite general fact that these solutions have an asymptotic expansion as a Gevrey series with respect to the variable xnprovided that we choose an integration path of rapid decay both for the function t−β−1exp (x1ta1+· · · +xntan) and for t−β−1exp (x1ta1+· · · +xn−1tan−1). At the beginning of Section 4 we prove that these expansions are Gevrey series of order less than or equal to an an−1. In the remainder of this Section we prove the main result of this paper about realisation of these Gevrey series as asymptotic expansion of integral solutions. First we treat in detail the case of dimension 2 and the last Subsection consists in using various restriction operations on the integral that are compared to the analogue described in Section 2 for the Gevrey series. In a short last section we deduce from the previous results an explicit description of the germ of the irregularity sheaf along Yat a generic point. We get a family of integral with asymptotic expansions that yields as expected a basis of the space of classes of Gevrey series solutions modulo convergent ones. 2. Gevrey solutions of hypergeometric systems In this Section we review some results on the construction of Gevrey series solutions of hypergeometric systems with respect to a coordinate hyperplane. We consider X=Cnand the hyperplane Ydefined by xn= 0. With the hypergeometric system MA(β) we associate the left coherent DX– module MA(β) := DX DXHA(β)which is called the analytic hypergeometric system associated with the pair (A, β). A germ fof the sheaf Od X|Yat a point (p, 0) ∈Yhas the form f=Pm≥0fm(x1, . . . , xn−1)xm n where all the fmare holomorphic functions in a common neighbourhood of p; in particular the restriction of OXto Y, denoted by OX|Y, is a subsheaf of Od X|Y. For any real number s, we consider the sheaf Od X|Y(s) of Gevrey series along Yof order less than or equal to sdefined as the subsheaf of Od X|Ywhose germs fat any (p, 0) ∈Ysatisfy the following convergence condition: X m≥0 fm(x1, . . . , xn−1) m!s−1xm n∈ OX|Y,(p,0). If s0< s then Od X|Y(s0)⊂ O d X|Y(s). If a germ fbelongs to Od X|Y(s) for some sbut f /∈ O d X|Y(s0) for all s0< s, we say that the index of the Gevrey series fis s. Let A= (a(1) · · · a(n)) be a full rank d×nmatrix with a(j)∈Zdfor j= 1, . . . , n. In [GZK89] and [SST], the authors associate with any vector v∈Cnsatisfying Av =βa series expression of the form (2.1) ϕA,β,v(x) := xvX u∈Nv Γ[v;u]xu where Nv={u∈kerZ(A)|nsupp(v+u) = nsupp(v)}, kerZ(A) = {u∈Zn:Au = 0}and nsupp(w) := {i∈ {1, . . . , n} | wi∈Z<0}is the negative support of w∈Cn. The coefficient Γ[v;u] equals [v]u− [v+u]u+where [v]u=Qi[vi]uiand [vi]ui=Qui j=1(vi−j+ 1) is the Pochhammer symbol for vi∈C,ui∈N. They call it the Γ-series associated with v. 4 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER We write ϕv=ϕA,β,v is no confusion is possible. It is proved in [SST, Proposition 3.4.13] that the formal expression ϕvis annihilated by the hypergeometric ideal HA(β) if and only if the negative support of vis minimal, which means that there is no u∈kerZ(A) with nsupp(v+u)(nsupp(v). When β∈Cdis very generic, that is when βis not in a countable union of Zariski closed sets, there is a basis of the Gevrey solutions space of MA(β) along Yat a generic point of Y, given by series ϕvfor suitable vectors v∈Cn, [F10, Th. 6.7]. In this article we restrict ourself to the case where A= (a1, . . . , an) is a row matrix with 0 < a1<· · · < an, (a1, . . . , an) coprime and n≥2. We recall here some results about the Gevrey series solutions of MA(β) as presented in [FC11b, Sections 4, 5]. Let us fix some notations. As before X=Cnand Y⊂Xdenotes the hyperplane defined by xn= 0. Let us write Z⊂Xthe hyperplane defined by xn−1= 0. When s < an an−1one sees from the results in [FC11b] that the set of Gevrey solutions in Od X|Y(s) is zero if β /∈NAand is a one dimensional space generated by a polynomial if β∈NA. This result follows also from what we show in this paper and we focus now on the case s≥an an−1. 2.1. Case a1= 1.Assume first n≥3. A basis of the free Z–module kerZ(A) is formed of the vectors {u(2), . . . , u(n)}where u(n−1) = (an−1,0,...,0,−1,0) and for i= 2, . . . , n,i6=n−1, we define u(i)= (−ai,0,...,0,1,0,...,0) where 1 is in the i-th component. For each m= (m2, . . . , mn)∈Zn−1we write u(m) = Pn i=2 miu(i) the corresponding element in kerZ(A). For j= 0, . . . , an−1−1 define vj= (j, 0,...,0,β−j an−1,0) ∈Cnand consider the associated Γ–series (2.2) ϕA,β,vj=xvjX m2,...,mn−1,mn≥0 j+an−1mn−1≥Pi6=n−1aimi Γ[vj;u(m)]xu(m). We write ϕ(j) A,β =ϕA,β,vjand also ϕ(j)=ϕA,β,vjif no confusion arises. By the choice of a basis of kerZ(A) that we make ϕ(j)is a series in xvjC[[x1, . . . , x−1 n−1, xn]][x−1 1]. Notice here that the summation in ϕ(j)is taken, according to (2.1), over the set Nv(j)={u(m)|nsupp(v(j)+ u(m)) = nsupp(v(j)}. This set Nv(j)is indexed by ({m∈Nn−1|j+an−1mn−1≥Pi6=n−1aimi}if β−j an−1/∈N {m∈Nn−1|j+an−1mn−1≥Pi6=n−1aimi,β−j an−1≥mn−1}if β−j an−1∈N. In this last case we can write β=j+han−1for unique 0 ≤j < an−1and h∈Nand then the series ϕ(j)is a polynomial since [h]mn−1= 0 = Γ[vj;u(m)] if mn−1≥h+ 1. In fact ϕ(j)is a polynomial if and only if we are in that case. The negative support of each vjis ∅if h∈Nand {n−1}if β−jis a negative multiple of an−1 and it is minimal in both cases. The series ϕ(j)is a polynomial in the first case and a solution of the ideal HA(β) in both. More precisely, Theorem 2.1. [FC11b, Th. 4.21, i)] Let A= (1, a2, . . . , an)∈Znwith 1< a2<· · · < an, Y= (xn= 0) ⊂Xand Z= (xn−1= 0) ⊂X. Then the set of germs at pof Gevrey series {ϕ(j)|j= 0, . . . , an−1−1}is a basis of HomDX(MA(β),Od X|Y(s))pfor all β∈C,p∈Y\Zand s≥an/an−1. GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 5 It is also useful to consider the immersion i:C3,→Xdefined by the equations x2=· · · =xn−2= 0 and the restriction ρ0with respect to this immersion (coordinates in C3are (x1, xn−1, xn)). The series ϕ(j) A,β(x1,0,...,0, xn−1, xn) equals precisely ϕ(j) (1,an−1,an),β(x1, xn−1, xn) so the restriction defines an isomorphism (2.3) HomDX(MA(β),Od X|Y(s)) ρ0 −−−−→ HomDC3(M(1,an−1,an)(β),O\ C3|Y1(s)) of the corresponding stalks at any point in Y1\Z1for all β∈Cand s∈Rwhere we denote Y1, Z1 the subspaces in C3with equations xn= 0 and xn−1= 0. Remark 2.2. So far we have assumed n≥3. The case n= 2 is special and will be treated now, following [FC11a]. We can drop in this case the assumption on a1and simply write a=a1, b =a2, with 1 ≤a < b and gcd(a, b)=1. The corresponding Γ–series have a slightly different shape (see [FC11a]): For j= 0, . . . , a −1, consider wj= (β−jb a, j)∈C2and (2.4) ψ(j) A,β =xwjX m≥0 [β−jb a]bm [am +j]am x−bm 1xam 2. We have the following Proposition 2.3. [FC11a, Prop. 5.3 and 5.4] Write X=C2,Y= (x2= 0) ⊂X. The set (of germs) of Gevrey series {ψ(j) A,β |j= 0, . . . , a−1}is a basis of the stalk of HomDX(MA(β),Od X|Y(s)) at any point in Y\ {(0,0)}, for any real number s≥b aand any β∈C. For a generic value of the parameter β,ϕ(j) (1,a,b)(x0, x1, x2) restrict by setting x0= 0 to another basis of the solution space in Proposition 2.3. The shape of the base change is a reindexation j→j0composed with a diagonal invertible matrix ϕ(j) (1,a,b)(0, x1, x2) = λjψ(j0) A,β. This fact and more, is explained in Proposition 2.7 1) below. 2.2. Case a1>1.Recall that X=Cn,Y⊂X(resp. Z⊂X) is defined by xn= 0 (resp. xn−1= 0). First of all, we follow [FC11b, Rk. 5.4] to prove the following equality Proposition 2.4. (2.5) dimCHomDX(MA(β),Od X|Y(s))p=an−1 if p∈Y\Zand s≥an an−1for any β∈C. Proof. We apply, among other results, Cauchy-Kovalevskaya’s Theorem for Gevrey series. We consider A0= (1, a1, . . . , an) and the hypergeometric system MA0(β) on X0:= Cn+1 with coordinates (x0, x1, . . . , xn). We denote by Y0⊂X0(resp. Z0⊂X0) the hyperplane xn= 0 (resp. xn−1= 0) and we identify X⊂X0with the hyperplane x0= 0. Notice that Y=Y0∩Xand Z=Z0∩X. By [CT03, Proposition 4.2] we can apply Cauchy-Kovalevskaya’s Theorem for Gevrey series solutions (see [LM02, Corollary 2.2.4]) to deduce that there exists a Cauchy-Kovalevskaya’s isomorphism CKs X0,X HomDX0(MA0(β),O\ X0|Y0(s))|X CKs X0,X −−−−−−→ HomDX(MA0(β)|X,Od X|Y(s)) 6 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER where MA0(β)|Xstands for the restriction in the category of D–modules. This is true for any s∈Rand for any βand even if gcd(a1, . . . , an)6= 1. We write CKs=CKs X0,X is no confusion is possible. The isomorphism CKsis induced by the inclusion X⊂X0, by the action of restriction on the modules involved. Theorem [FC11b, Th. 5.1] states that for any β∈Cthere exists β0∈Csuch that the restriction MA0(β)|Xis isomorphic to the hypergeometric DX–module MA(β0). Moreover, by the same Theorem [FC11b, Th. 5.1] one can take β0=βfor all but finitely many β. We denote by sp(A) the finite set of β∈Csuch that β06=β. In particular, the isomorphism CKsand Theorem 2.1 prove equality (2.5) for p∈Y\Z, s≥an an−1and β6∈ sp(A). Assume now β∗∈sp(A). We can take β=β∗+A0γ0=β∗+Aγ for a suitable γ0= (0, γ)∈Nn+1 ∪ (−N)n+1 in such a way that β6∈ sp(A) and so the corresponding morphism CKsis an isomorphism. By using [S01, Th. 2.1], [B11, Th. 6.5] one has that the morphism ·∂γ:MA(β∗)→ MA(β) if γ∈Nn(resp. ·∂−γ:MA(β)→ MA(β∗) if γ∈(−N)n) is an isomorphism. This proves equality (2.5) for β∗∈sp(A).  Once the equality (2.5) is established we give a description of a basis of the solution space of MA(β) in Od X|Y(s)). We use the restriction morphism (2.6) HomDX0(MA0(β),O\ X0|Y0(s))|X ρs X0,X −−−−−→ HomDX(MA(β),Od X|Y(s)) for β∈Cand s∈R. This morphism is well defined and it is induced by the restriction to x0= 0. It is useful to write x0= (x0, x) and x= (x1, . . . , xn). Then ρs X0,X(ϕ(x0)) = ϕ(0, x), since if ϕ(x0) is a solution in the first space then ϕ(0, x) is a solution in the second one. Notice that it is an approach of restriction that is different from the one by the CK’s. We now consider the basis {ϕ(j) A0,β(x0, x)}an−1−1 j=0 of germs (at a point in Y\Z⊂Y0\Z0) of Gevrey series solutions (of order ≤s) of MA0(β) described in Theorem 2.1.We simply write ϕ(j)=ϕ(j) A0,β. The terms in x −β+j an−1 n−1ϕ(j)have the form Γ[v(j);u(m)]xj 0(x0)u(m)= Γ[v(j);u(m)]xj+an−1mn−1−Pi6=n−1aimi 0xm1 1· · · xmn−2 n−2x−mn−1 n−1xmn n where u(m) is a general element of kerZ(A0)⊂Zn+1. The summation in ϕ(j)(see (2.2)) is taken over the set {m∈Nn|j+an−1mn−1≥Pi6=n−1aimi}. It is useful to write the formal expansion of ϕ(j)(0, x). According to what is said before we have (2.7) ϕ(j)(0, x) = x β−j an−1 n−1X m1,...,mn−1,mn≥0 j+an−1mn−1=Pi6=n−1aimi [β−j an−1]mn−1j! m1!· · · mn−2!mn!xm1 1· · · xmn−2 n−2x−mn−1 n−1xmn n. Notice that the family of non zero ϕ(j)(0, x) is C–linearly independent because their supports are pairwise disjoint. This, and equality (2.5), proves the following (see [FC11b, Remark 5.4]) Theorem 2.5. Assume that ϕ(j)(0, x)is non zero for j= 0, . . . , an−1−1. Then the set (of germs of) Gevrey series {ϕ(j)(0, x)|j= 0, . . . , an−1−1}is a basis of the stalk of HomDX(MA(β),Od X|Y(s)) at any point in Y\Zfor s≥an an−1. GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 7 Remark 2.6. Assume now that for some j= 0, . . . , an−1−1 one has ϕ(j)(0, x) = 0. By [FC11b, Rk. 5.4] this condition happens if and only if β∈N\NA. Furthermore this jis then unique for a fixed βand the image of the morphism ρs X0,X has codimension 1 in the stalk of HomDX(MA(β),Od X|Y(s)) at any point in Y\Zand for all s≥an an−1. We will show in Theorem 4.8 how to describe a basis of this last solution space for all β∈C. 2.3. The restriction to x0= 0 for A0= (1, ka, kb).Let us consider the morphism ρs X0,X when X= (x0= 0) ⊂X0=C3and the matrix A0= (1, ka, kb) with 1 ≤a<b, 1 < ka and gcd(a, b) = 1. The morphism ρs X0,X, as defined in (2.6), sends solutions of MA0(β) to solutions of M(ka,kb)(β) and this last DX–module is isomorphic to M(a,b)(β k). In particular if k > 1, ρs X0,X is not an isomorphism, since the corresponding solutions spaces have dimensions ka and arespectively. Below we describe in detail a different but related morphism involving the restrictions of the derivatives up to order k−1 with respect to the variable x0. So, instead of considering A= (ka, kb) as before it is better to write A= (a, b). Coordinates in X0are (x0, x1, x2). For j= 0, . . . , ka −1 let us write v(j)= (j, β−j ka ,0) and ϕ(j) A0,β =x β−j ka 1X m1,m2≥0 j+kam1≥kbm2 [β−j ka ]m1j! m2!(j+kam1−kbm2)!xj+kam1−kbm2 0x−m1 1xm2 2. Notice that if a series ϕ(as for example ϕ(j) A0,β) is a solution of MA0(β) then ∂`ϕ ∂x` 0x0=0 is a solution of the system MA(β−` k) for all `≥0. We are going to compare this solution, for ϕ=ϕ(j) A0,β, to the usual Gamma series solutions ψ(j0) A,β0(x1, x2), see (2.4), of this last system MA(β0) for β0=β−` k. We consider the following C–linear map (2.8) HomDX0(MA0(β),O\ X0|Y0(s))|X $s β −−−−−→ k−1 M `=0 HomDXMAβ−` k,Od X|Y(s) which maps ϕto the vector ∂`ϕ ∂x` 0(0, x1, x2)k−1 `=0 . Here Y0= (x2= 0) ⊂X0and Y=X∩Y0. Notice that the morphism $s βis well defined and that it coincides with the morphism (2.6) when k= 1. The following Proposition generalizes to an arbitrary kwhat is already proved for k= 1 in Theorem 2.5 and in Remark 2.6 : Proposition 2.7. 1) Assume that β6∈ Nor β∈Sk−1 r=0(r+kNA). Then the C–linear map $s βis an isomorphism of vector spaces. 2) Assume β∈Sk−1 r=0(r+k(N\NA)) and consider the integers j0, r0, q0uniquely determined by 0≤j0< ka,0≤r0< k and β=j0+kah =r0+kq0+kah. Then the image of the map $s βis the codimension 1subspace, generated by all the (0,...,0, ψ(j0) A, β−r k ,0,...,0), with a non zero term in position r, and (j0, r)6= (j0 0, r0)for j0 0∈ {0, . . . , a −1}uniquely determined by bj0 0≡q0(mod a). Proof. We consider the morphism $s βinduced on the stalks at points in Y\Z,Z= (x1= 0) ⊂X and s≥b a. Let us fix j= 0, . . . , ka −1 and write j=kq +rfor unique 0 ≤q < a and 0 ≤r < k. 8 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER So we have β−j ka =β−kq −r ka =β−r ka −q a= β−r k−q a. The general exponent of x0in ϕ(j) A0,β is j+k(am1−bm2) = r+k(q+am1−bm2). An exponent of x1in ∂`ϕ(j) A0,β ∂x` 0x0=0 for 0 ≤`<k, can only come from a term in ϕ(j) A0,β for which the underlying exponent of x0is `. Therefore the only pairs (m1, m2)∈N2that may appear are those satisfying the relation : (2.9) j+k(am1−bm2)−`=k(q+am1−bm2) + r−`= 0. This cannot happen if `6=rwhich means that ∂`ϕ(j) A0,β ∂x` 0x0=0 = 0 if `6=r. The equality (2.9) is equivalent to q=bm2−am1and `=r. For any (m1, m2)∈N2satisfying (2.9), we write m2=am+j0with 0 ≤j0< a. We notice that the integer j0depends only on q, hence on jbecause q=b(am+j0)−am1so that adivides bj0−q. For the rest of the proof we denote it j0=p(j). By a straightforward calculation the general exponent of x1in ∂rϕ(j) A0,β ∂xr 0x0=0 is β−j ka −m1=β−r ka −bp(j) a−bm which equals the general exponent of x1in ψ(p(j)) A,β0by (2.4) applied to β0=β−r k. The series ϕ(j) A0,β and ψ(p(j)) A,β0are indexed respectively by (m1, m2) and m. Their terms are in a 1-1 correspondence through the relations m1=bm +bp(j)−q aand m2=am +p(j). Now we compare the corresponding coefficients in ∂rϕ(j) A0,β ∂xr 0x0=0 and in ψ(p(j)) A,β0. The quotient of these two coefficients is well defined, for all m≥0, when β /∈Nand it is (2.10) [β−kq−r ka ]m1(kq +r)! m2!! [β0−p(j)b a]bmp(j)! (am +p(j))! !−1 =Γ(z−bm +m1)(kq +r)! Γ(z)p(j)! where z−1 = β0−p(j)b a=β−r ka −p(j)b a. This shows that this quotient is independent of the term chosen, because the integer bm −m1= q−bp(j) adoes not depend on m, m1. So, if β6∈ Nwe have found a constant λj∈C∗such that ∂rϕ(j) A0,β ∂xr 0x0=0 =λjψ(p(j)) A, β−r k and ∂`ϕ(j) A0,β ∂x` 0x0=0 = 0 if `6=r. When β∈N, we write β=j0+kah =r0+kq0+kah, with unique 0 ≤r0< k and 0 ≤q0< a. If j6=j0=r0+kq0the quotient in (2.10) is still well defined for all mand the relation λj6= 0 still valid. If β∈r0+k(N\N(a, b)), the quotient is still well defined but we have λj0= 0 by Remark 2.6 since ϕ(j0) A0,β is a polynomial solution of MA0(β) and MAβ−r0 khas no non zero polynomial solution for β−r0 k/∈N(a, b). Assume now β−r0 k=ha +q0∈N(a, b) so that ha +q0=ua +vb for some u, v ∈N. We can write in a unique way q0=−n1a+n2bwith 0 ≤n2< a. One has (h−n1)a+n2b=ua +vb which forces GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 9 v≥n2and h−n1−u≥0. In particular, h≥n1and [h]n16= 0 and so λj06= 0. This proves 1) and 2) taking j0 0:= p(j0).  3. Hypergeometric integral of exponential type With the pair (A, β) one associates the following integral (called hypergeometric integral of exponential type): Iγ(A, β;x) = I(β;x) := Zγ t−β−1exp n X j=1 xjtaj!dt where A= (a1, . . . , an) and γis a cycle in the rapid decay homology with closed support of M. Hien. We also write Iγ(β;x) = I(β;x) if there is no possible confusion on the matrix A. The integral I(β;x) satisfies the equality P(I(β;x)) = 0 for any Pin the hypergeometric ideal HA(β). So we consider I(β;x) as a solution of MA(β). Our goal is to give an asymptotic expansion of I(β;x) as Gevrey series along the coordinate hyperplane appearing in the singular support of MA(β). We are going to prove in this case that all the Gevrey series solutions can be obtained as an asymptotic expansion of such an integral. In the one row matrix case the rapid decay cycles are easy to describe and we prove first an asymptotic expansion statement. In the following proposition we consider a path γ:R→Csuch that the arguments of xntanand of xn−1tan−1both have limits in the open interval ]π 2,3π 2[ when t→+∞or t→ −∞. For a fixed γthis condition remains valid in some open sectors in the spaces C∗for the variables xn−1and xn. These paths are exactly the rapid decay cycles for the function t→t−β−1exp Pn j=1 xjtajand for its restriction to the hyperplane xn= 0. Proposition 3.1. The integral depending on x= (x1, . . . , xn): I(β;x) = Zγ t−β−1exp n X j=1 xjtaj!dt admits an asymptotic expansion P∞ k=0 ck(x1, . . . , xn−1)xk nfor xntending to zero whose coefficients are (3.1) ck(x1, . . . , xn−1) = 1 k!Zγ t−β−1+ankexp (x1ta1+· · · +xn−1tan−1)dt. For a fixed cycle γthis expansion is valid in a product Cn−2×Sn−1×Sninvolving open sectors Si in C∗. Proof. By developing the exponential exntanwe may write I(β;x) = Zγ t−β−1exp (x1ta1+· · · +xn−1tan−1) ∞ X k=0 (xntan)k k!!dt and we get (3.2) I(β;x1, . . . , xn) = N X k=0 xk n k!Zγ tank−β−1exp (x1ta1+· · · +xn−1tan−1)dt +RN(β;x1, . . . , xn) 16 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER and one show by elementary calculations det M= 1 1 1 . . . 1 1q0q2 0. . . qa 0 . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . 1qa−1q2 a−1. . . qa a−1  = a−1 Y i=0 (qi−1) Y i<j (qj−qi)6= 0 where the last inequality follows from qj=e2iπ jb−β a, with β /∈Zand a, b co-prime.  Theorem 4.7. For any β∈Cand for every Gevrey series ϕ∈ O d X|Y(s),s≥b a, which is a solution of the hypergeometric system MA(β), there is an holomorphic solution defined in a product of sectors, which is a neighbourhood of the product of the real negative axes, and which admits ϕ as an asymptotic expansion. All these solutions can be described as an integral of the function t−β−1exp(x1ta+x2tb)along a rapid decay cycle when β /∈Z\N(a, b). When β∈Z\N(a, b), the above integral solutions span a codimension one subspace and there is a one dimensional supplementary space obtained by expanding an integral along [0,+∞[. Proof. When β /∈Z, Proposition 4.6 gives a complete proof of the statement. When β∈Zwe write β=j0b+aq, 0 ≤j0< a. The proof of the same Proposition solves the case of the Gevrey series ψ(j)for j6=j0. Indeed the matrix Min this proposition has its last column equal to zero and the row corresponding to ψ(j0)is zero too. We find that the matrix Mis of rank exactly a−1 so that all ψ(j)for j6=j0are obtained as integrals on rapid decay cycles. The non obtained Gevrey series is: ψ(j0)=X m≥0 [q]bm [am +j0]am xq−bm 1xam+j0 2. It is a polynomial if and only if q≥0, that is when β∈N(a, b). We notice that it is exactly the case where the integral along a circle of radius  > 0 centered at the origin which is equal to ZCb t−β−1ex1ta+x2tbdt = 2πi X `1a+`2b=β x`1 1x`2 2 `1!`2! is non zero. Since this is a polynomial solution we are done in this case. Finally when β∈Z\N(a, b), that is when q < 0, we notice that there is an integral holomorphic solution given by the formula: (4.7) Jβ(x1, x2) = Z+∞ 0 t−β−1ex1ta+x2tb−Pβ(x1, x2, t)dt where Pβis zero if β < 0, and otherwise is the Taylor polynomial of degree ≤βfor t→ex1ta+x2tb: Pβ(x1, x2, t) = X `1a+`2b≤β x`1 1x`2 2t`1a+`2b `1!`2!. In fact degtPβ< β, by the condition β /∈N(a, b). This yields the convergence of Jβat +∞. The fact that Jβis a solution of the system HA(β) is completely similar to the proof in [A94, Section 2] for the case of rapid decay cycles. The reason is first that since ∂b 1−∂a 2annihilates ex1ta+x2tb, it annihilates as well the coefficients of all the monomials tjin its power expansion and GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 17 hence also the polynomial Pβ(x1, x2, t). Concerning the Euler operator χ=ax1∂1+bx2∂2we just have to notice that the relation for the integrand which leads to the proof is still valid : χex1ta+x2tb−Pβ(x1, x2, t)=t∂tex1ta+x2tb−Pβ(x1, x2, t). In order to end the proof of Theorem 4.7 we just have to check that Jβadmits under the same conditions as in Section 4.1 an asymptotic expansion (of Gevrey order less than or equal to b a) which must then be a linear combination Pµjψ(j)of all the ψ(j). We rewrite it as follows Jβ(x1, x2) = Z+∞ 0 ∞ X k=0 t−β−1+bk ex1ta−X `a≤β−kb x` 1t`a `!!xk 2 k!!dt and this leads to the existence of an asymptotic expansion completely similar to the one given in the general case, but with a coefficient of xk 2given by : (4.8) ck(x1) = 1 k!Z+∞ 0 t−β−1+bk ex1ta−X `a≤β−kb x` 1t`a `!!dt which is again a convergent integral. The change of variable s=−x1tafor x1∈R<0works exactly in the same way as in Remark 4.4 and Proposition 4.5. In fact as soon as kis large enough the correcting term P`a≤β−kb x` 1t`a `!is zero and we find explicitly : ck(x1) = 1 ak!e−iπ β−bk aΓbk −β ax β−kb a 1. Considering the coefficients cam+j0(x1) for mlarge enough, we see that the coefficient µj0of ψ(j0) A,β is non zero in the asymptotic expansion of Jβas expected.  4.2. A basis of Gevrey asymptotic expansions. Let A= (a1, . . . , an) with integers 0 < a1< · · · < anand gcd(a1, . . . , an) = 1. By a rapid decay cycle we always mean in this section a rapid decay cycle for the exponentials of both polynomials P1≤j≤nxjtajand P1≤j≤n−1xjtaj. In Propositions 3.1 and 4.1 we prove that for any β∈Cand any rapid decay 1-cycle γthere is an asymptotic expansion of the integral Iγ(A, β, x). We denote it : Φγ(A, β;x)=a.e.(Iγ(A, β, x)) = a.e. Zγ t−β−1exp n X j=1 xjtaj!dt!. We also prove that this asymptotic expansion Φγ(A, β;x) is a germ of Gevrey series in Od X|Y(s) at any point in Y\Zfor all s≥an an−1. Let us consider a vector space Econsisting in formal linear combinations of geometric cycles of the above type. We get a map γGE//Φγfrom Eto the space of Gevrey series solutions of the system. When β∈Z\NA, and restricting to a product of sectors in the variables xn−1, xncentered on half lines whose respective arguments are (π−an−1θ and π−anθ), we may as in the previous Section define an asymptotic expansion for the integral Jβ(x1, . . . , xn) = Zeiθ·[0,+∞[ t−β−1ex1ta1+···+xntan−Pβ(x1, . . . , xn, t)dt. Here Pβ(x1, . . . , xn, t) is again the Taylor polynomial of degree βfor t→exp (x1ta1+· · · +xntan). When β∈Z\NAwe still denote Φγ(A, β;x) such an asymptotic expansion. By extension we still call a vector space of cycles the formal direct sum of a one dimensional space C·eiθ ·[0,+∞[ and a subspace made of rapid decay cycles. 18 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER The goal of the remaining part of this Section is to prove the following generalisation of the surjectivity statement for GEthat follows directly from Theorem 4.7. Theorem 4.8. There exists an an−1dimensional vector space Eof cycles such that the map GE from Eto the set of germs of Gevrey asymptotic expansions is an isomorphism onto the stalk of HomDXMA(β),Od X|Y(s)at any point in Y\Zfor s≥an an−1. Remark 4.9. 1) By [FC11b, Remark 4.12] it is enough to prove the Theorem for points of the form (0,...,0, ε, 0) ∈Xwith ε6= 0. 2) In the proof of this Theorem we will see that for β /∈Z\NAthe space Eis a space of rapid decay cycles. When β∈Z\NAthere exists a unique eiθ ·[0,+∞[ which spans a one dimensional complement in Eto a space of rapid decay cycles. 3) In the statement of Theorem 4.8 it must be understood that a Gevrey expansion can be obtained along any half line eiθ ·[0,+∞[ in the variable xn. The space Edepends on the arguments chosen for xn−1, xn. In the proof of Theorem 4.8 we first focus on products of sectors in the variables xn−1, xnthat are centered along the real negative axes, while the other variables x1, . . . , xn−2are arbitrary in C. The general statement follows easily from this particular case by considering actions of roots of unity on the cycles and by controlling the width of the sectors of validity of the asymptotic expansions. Corollary 4.10. [of Theorem 4.7] Theorem 4.8 holds for n= 2 and a pair of sectors centered on the real negative axis for each of the variables x1, x2. In order to simply notations we denote Sol(M) = HomD(M,Od X|Y(an an−1)) the sheaf of Gevrey solutions of order less than or equal to an an−1of a holonomic DX–module Mon X=Cnfor Y= (xn= 0) ⊂X. Remark 4.11. By Corollary 4.10, for all β∈Cand for each r= 0, . . . , k −1 there exists a family γ1,r, . . . , γa,r of cycles such that the family Φγ1,r (a, b),β−r k;x1, x2,...,Φγa,r (a, b),β−r k;x1, x2 is a basis of the stalk of the solution space Sol(M(a,b)(β−r k)) at any point in Y1\Z1. Here Y1⊂C2 (resp. Z1⊂C2) is the line x2= 0 (resp. x1= 0). In the statement of the following Proposition we use the morphism $β:= $s β, for s=an an−1, defined in (2.8). Recall that, according to Proposition 2.7 in Section 2, $βis an isomorphism if β /∈N\Sk−1 r=0(r+kN(a, b)). But if there exists 0 ≤r0< k (necessarily unique) such that β−r0 k∈Z\N(a, b) then one of the cycles γτ0,r0is [0,+∞[. All the other cycles γτ,r are in the set of rapid decay ones and their asymptotic expansions span the image of $βwhich contains no non zero polynomial. Proposition 4.12. Theorem 4.8 holds for the matrix (1, ka, kb)with gcd(a, b)=1and k≥1. There exists a set of cycles {eγτ,r |1≤τ≤a, 0≤r < k}, such that the asymptotic expansions Φeγτ,r ((1, ka, kb), β)span the space of solutions Sol(M(1,ka,kb)(β)).More precisely : If $βis an isomorphism, the image of the germ of Φeγτ,r ((1, ka, kb), β)by the morphism $βequals 0,· · · ,0,1 kΦγτ,r (a, b),β−r k,0,· · · ,0. GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 19 If $βis not an isomorphism, the same is true for all the pairs (τ, r)such that Φγτ,r is in the image of $β. This exclude exactly one pair (τ0, r0)characterised by β=r0+kq0, and Φτ0,r0not in the image of $β. Furthermore Φeγτ0,r0((1, ka, kb), β)is a polynomial. Proof. We consider the germs at a point (0, ε, 0) with ε6= 0 and the stalk of the direct sum at (ε, 0). We write B0= (1, ka, kb) and B= (a, b), and (x0, x1, x2) for coordinates in C3. The proof here depends heavily on Proposition 2.7 in Section 2. Let us assume first that γτ,r is one of the rapid decay cycles among those considered in Remark 4.11. They are relative to the matrix Band β−r kand this excludes [0,+∞[, when β∈r0+k(Z\N(a, b)) for some r0∈ {0, . . . , k −1}. Denote it for short γ:= γτ,r and choose a kth root eγin C∗: eγ(t)k=γ(t). The cycle eγis of rapid decay with respect to B0and we develop the integral Ieγ=Ieγ(x0, x1, x2) = Ieγ(B0, β;x0, x1, x2) = Zeγ t−β−1exp(x0t+x1tka +x2tkb)dt as Ieγ(x0, x1, x2) = k−1 X `=0 x` 0J` eγ(xk 0, x1, x2). Notice that ∂`Ieγ ∂x` 0(0, x1, x2) = `!J` eγ(0, x1, x2) and the change of variables s=tkshows that : IγB, β−r k;x1, x2:= Zγ s−β−r kexp(x1sa+x2sb)ds s =Zeγ t−β+rexp(x1tka +x2tkb)kdt t=k∂rIeγ ∂xr 0 (0, x1, x2) Let ω=e2iπ kand for ν= 0, . . . , k −1, eγ(ν):= ω−νeγ. We can write Ieγ(x0, x1, x2) = ω−νβ Zeγ(ν) s−β−1exp(x0ωνs+x1ska +x2skb)ds =ω−νβIeγ(ν)(x0ων, x1, x2) and then Ieγ(ν)(x0, x1, x2) = ωνβIeγ(x0ω−ν, x1, x2) = ωνβ k−1 X `=0 ω−ν`x` 0J` eγ(xk 0, x1, x2). The matrix (ω−ν`)0≤ν,`≤k−1being invertible we can write (4.9) x` 0J` eγ(xk 0, x1, x2) = k−1 X ν=0 λ`,νω−νβIeγ(ν)(x0, x1, x2) for some λ`,ν ∈C. Now the cycle eγτ,r := Pk−1 ν=0 λr,νω−νβeγ(ν)yields the expected result for γ=γτ,r :    ∂`Ieγτ,r ∂x` 0(0, x1, x2) = 0 if `6=r ∂rIeγτ,r ∂xr 0(0, x1, x2) = r!Jr eγτ,r (0, x1, x2) = 1 kIγB, β−r k;x1, x2. Consider the non rapid decay case γτ0,r0= [0,+∞[. This happens when β∈Z\Sk−1 r=0(r+kN(a, b)), β=β0k+r0for unique integers β0and 0 ≤r0< k. We know by Theorem 4.7 that Φτ0,r0is the asymptotic expansion attached to the integral : Jβ−r0 k,B(x1, x2) = Z+∞ 0 s−β−r0 k(exp(x1sa+x2sb)−Pβ−r0 k (x1, x2, s))ds s 20 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER When $βis an isomorphism, we consider the expression Jβ,B0(x0, x1, x2) = Z+∞ 0 t−β−1(exp(x0t+x1tka +x2tkb)−Pβ(x0, x1, x2, t))dt with notations analogous to those used in the proof of Theorem 4.7. It can be proved as in Theorem 4.7 that Jβ,B0is a holomorphic solution of the system MB0(β) and that it admits an asymptotic expansion which is of Gevrey order less than or equal to b athat we denote Φ[0,+∞[. A straightforward calculation, with the change of variables s=tkshows that: ∂r0Jβ,B0 ∂xr0 0 (0, x1, x2) = Jβ−r0 k,B(x1, x2) so that the r0-component of $β(Φ[0,+∞[), is equal to Φτ0,r0. Using the already determined asymptotic expansions for all others Φeγp,r we get the expected result with a uniquely determined linear combination eγτ0,r0= [0,+∞[−P(τ,r)6=(τ0,r0)cτ,reγτ,r. If β∈N, the integral along the cycle Cb(see notations in Subsection 4.1) ZCb t−β−1exp(x0t+x1tka +x2tkb)dt is a polynomial. Therefore if β∈r0+k(N\N(a, b)) for some r0∈ {0...,k−1}, its image by $β is zero and it is the missing asymptotic expansion in the solution space Sol(M(1,ka,kb)(β)) by the final part of Remark 4.11. This finishes the proof.  Proof. [of Theorem 4.8, general case]. We may assume n≥3. We simply write ka =an−1, kb =an for some integer k≥1 and gcd(a, b) = 1. We first assume a1>1 and denote as usual A0= (1, a1,· · · , an). We also denote B0= (1, an−1, an) = (1, ka, kb) and B= (a, b). Let us consider the following diagram Sol(MA0(β))p ρ0 // ρ  Sol(MB0(β))p Sol(MA(β))p. Let us explain the restriction morphisms in the above diagram: ρ0is the restriction defined as ρ0(f(x0, x1, . . . , xn)) = f(x0,0,...,0, xn−1, xn). Similarly ρ(f(x0, x1, . . . , xn)) = f(0, x1, . . . , xn). We consider a point p= (0, . . . , ε, 0) ∈Cn+1 with ε6= 0 and we also denote pthe image of this point in the different considered spaces. The morphism ρ0is an isomorphism for any β∈C, see Subsection 2.1. The morphism ρis an isomorphism if β∈N\NA, see Theorem 2.5 and Remark 2.6. Let us consider the set of asymptotic expansions {Φeγτ,r (A0, β)|1≤τ≤a, 0≤r < k} where eγτ,r is the cycle built in the proof of Proposition 4.12. The image by ρ0of Φeγτ,r (A0, β) is just Φeγτ,r (B0, β). This proves the theorem for A0and then for Aif β6∈ N\NA(because in this case ρ is an isomorphism and the image of Φeγτ,r (A0, β) is precisely Φeγτ,r (A, β)). GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 21 If β∈N\NAthen we have again ρ(Φeγτ,r (A0, β)) = Φeγτ,r (A, β) for all (τ, r)6= (τ0, r0). The integral (4.10) Jβ:= Z+∞ 0 t−β−1 exp n X i=1 xitai!−Pβ(x1, . . . , xn, t)!dt defines a holomorphic function in a domain Cn−2×Sn−1×Snwhere Siis a sector in Cwhich is a neighbourhood of the real negative axis for i=n−1, n. Here Pβis the Taylor polynomial, for the exponential, of degree ≤βin t. As usual we consider, for kbig enough, (4.11) ck(x1, . . . , xn−1) = 1 k!Z+∞ 0 t−β−1+ankexp n−1 X i=1 xitai!dt the coefficient of xk nin the expansion of Jβ. By developing we get ck(x1, . . . , xn−1) = X m1,...,mn−2≥0 cm1,...,mn−2,k(xn−1)xm1 1· · · xmn−2 n−2 where, writing mn=kbig enough, cm1,...,mn−2,mn(xn−1) = 1 m1!· · · mn−2!mn!Z+∞ 0 t−β−1+Pi6=n−1aimiexn−1tan−1dt. Up to a scalar multiple this last integral equals x β−Pi6=n−1aimi an−1 n−1Γ−β+Pi6=n−1aimi an−1. The condition β∈N\NAimplies that the argument of the Gamma factor is never a non-positive integer. Writing β=qan−1+j0with 0 ≤j0< an−1and choosing m1, . . . , mn−2, mn−1, mn≥0 such that j0+an−1mn−1=Pi6=n−1aimiwe see that the corresponding exponent of xn−1in the expansion of Jβis β−Pi6=n−1aimi an−1=q−mn−1which is a negative integer if mn−1is large enough. Moreover, the asymptotic expansion of Jβis a Gevrey series solution of MA(β) of order less than or equal to an an−1which is linearly independent of the set {ϕ(j) A0,β(0, x)|j6=j0}. This finishes the proof of the theorem for Aif a1>1. If finally a1= 1 then we apply previous discussion by using the restriction to the case (1, an−1, an).  Up to now we have considered only asymptotic expansions in a neighbourhood of the real negative axes for xn−1, xn. Looking at the reductions that we have carried out we see that it is sufficient to check the general statement about the arguments when n= 2. We set A= (a, b). If we consider x2=y2eiθ of argument θand changing all the cycles γinto eiπ−θ b·γ, we are reduced to the real negative case for x2. Now we conclude with the following : Lemma 4.13. Theorem 4.8 holds for n= 2 in a product of sectors near the real negative axis for x2and an arbitrary argument for x1. Proof. Changing the cycles γto e2ipπ/b ·γpreserves the hypothesis for x2and modify the argument of x1by a factor e2ipaπ/b. By varying p, and because (a, b) = 1 we get all the b-roots of unity. It is therefore sufficient to check that the asymptotic expansion found in Subsection 4.1 is valid in sectors around the real negative axis whose union for a given `has a width strictly greater than 2π b. This follows from a careful inspection of the proof of the enlargement statement in Proposition 4.5.  22 FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ AND MICHEL GRANGER 5. Gevrey solutions modulo convergent solutions We can also give a description of the stalk of the solution space HomDX(MA(β),QY(s)) at any point of Y\Zwhere, for s≥1, QY(s) is the quotient of Od X|Y(s) by OX|Y. By [FC11b, Th. 5.3] this space is just (0) if 1 ≤s < an an−1and has dimension an−1if s≥an an−1. We will assume in this Section that s≥an an−1. Let φ(t) a C∞function with compact support locally constant with value 1 near the origin. We consider the following integral, see (4.10): (5.1) Jφ,β(x) = Jφ,β(x1, . . . , xn) := Z+∞ 0 t−β−1ex1ta1+···+xntan−Pβ(x, tφ(t))dt where Pβ(x, t) is the Taylor polynomial of the exponential of degree in tless than or equal to β. We write Jφ,β(A;x) if we want to emphasize the dependence of this integral on the matrix A. This integral defines a holomorphic function in Cn−2×Sn−1×Snfor some open sectors Si⊂C, i=n−1, n, each of them containing the real negative axis. In general this integral is not a solution of the hypergeometric system MA(β) but it is a solution modulo convergent power series. Now we come back to the Gevrey solutions modulo convergent ones. We will treat the case A= (a, b) first. According to Corollary 4.10, for β6∈ Zthe family of asymptotic expansions {Φτ(A, β) := Φγτ(A, β), τ = 1, . . . , a}is a basis of Gevrey solutions of MA(β) in Od X|Y(s). For β6∈ Z, the family {ψ(j) A,β |j= 0, . . . , a−1}is also a basis of Gevrey solutions and their classes modulo convergent series form a basis of HomDX(MA(β),QY(s)) [FC11a, Th. 5.9 (i)]. Then, the classes of {Φτ(A, β), τ = 1, . . . , a}modulo convergent series forms a basis of HomDX(MA(β),QY(s)). Previous situation is still valid when β∈Z\N(a, b), taking into account that we denote Φa(A, β) the asymptotic expansion of an integral Jβ(x1, x2) over γ= [0,+∞[, see Theorem 4.7. For p=b the integer `is just aand the integral along the cycle Cbis zero. As explained in the proof of Theorem 4.7 the asymptotic expansion of Jβ(x1, x2) replace the missing integral along rapid decay cycles. Assume now β∈N(a, b) and write β=j0b+qa with 0 ≤j0< a and q≥0. We know that the Gamma series ψ(j0) A,β is a polynomial and then its class modulo convergent series is zero. We denote by Φa(A, β) the asymptotic expansion of Jφ,β(x1, x2), see (5.1). The coefficient ck(x1) of this expansion is, for kbig enough, exactly the same as in (4.8). In particular, the exponent of x1in cam+j0(x1) for mbig enough is the negative integer q−bm. This proves that the family {Φτ(A, β), τ = 1, . . . , a}is still linearly independent modulo convergent power series. Hence, it defines a basis of HomDX(MA(β),QY(s)). We treat now the case A= (a1, . . . , an) and n≥3. We write as usual an−1=ka, an=kb for k≥1 and gcd(a, b) = 1. According to Theorem 4.8, for β6∈ Nthe family of asymptotic expansions {Φeγτ,r (A, β),1≤τ≤a, 0≤r < k}is a basis of Gevrey solutions of MA(β) in Od X|Y(s). For β6∈ N, the family {eϕ(j) A,β := ϕ(j) A0,β(0, x)|j= 0, . . . , an−1−1}is also a basis of Gevrey solutions and their classes modulo convergent series form a basis of HomDX(MA(β),QY(s)) [FC11b, Th. 5.5, (i)]. Then, the classes of {Φeγτ,r (A, β),1≤τ≤a, 0≤r < k}modulo convergent series form a basis of HomDX(MA(β),QY(s)). Previous situation is still valid when β∈N\NA, taking into account that we denoted Φeγτ0,r0(A, β) the asymptotic expansion of an integral Jβ(x) over [0,+∞[, see (4.10) and we considered this asymptotic expansion as the generator of a complement space of {eϕ(j) A,β, j = 0, . . . , an−1−1}. Assume now β∈NA. We denote by Φeγτ0,r0(A, β) the asymptotic expansion of Jφ,β(x), see (5.1), with respect to xn. The coefficient ck(x1, . . . , xn−1) of this expansion is, for kbig enough, exactly GEVREY EXPANSIONS OF HYPERGEOMETRIC INTEGRALS I 23 the same as in (4.11). 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