数理解析研究所講究録1090 巻 1999 年 100-109

100

Irregularities on hyperlanes of holonomic$D$ -module (especially defined by confluent
hypergeometric partial differential equations)真島 秀行
お茶の水女子大学理学部数学科(Hideyuki Majima, Ochanomizu University)

1

Intoduction

In this expository paper, I will explain the irregularity at a singular point of differential
equation. At first, I will give you a review of study on ordinary linear differential equations. Secondly, I will talk about holonomic -modules, especially, Humbert confluent
hypergeometric differential modules in $m$ variables.
$D$2

Irregularity of holonomic $D$ -module defined by an
ordinary differential operator.

Consider a linear ordinary differential operator with coefficients in holomorphic functions at the origin in the Riemann Sphere:$Pu=( \sum_{i=0}^{m}a_{i}(X)(d/dx)i)u$.

is supposed not to be identically zero. Let $O$ and be the ring of convergent
where
power-series and the ring of formal power-series in , respectively. Then, we see thefollowing isomorphism of linear spaces due to Deligne (cf. [23], etc.) :$\hat{O}$$a_{m}$$x$$H^{1}(S^{1}, \mathcal{K}er(P:A\mathrm{o}))\simeq \mathrm{K}\mathrm{e}\mathrm{r}(P;\hat{\mathcal{O}}/O)$,

for, from the existence theorem of asymptotic solutions due to Hukuhara (cf. [26]) (andother many contributers), we have the short exact sequence$\mathrm{O}arrow \mathcal{K}er(P:A\mathrm{o})arrow A_{0}-^{P}A_{0}arrow 0$,

from which, we get the exact sequence,$0arrow H^{1}(S^{1}, \mathcal{K}er(P:A0))arrow H^{1}(S^{1}, A\mathrm{o})(=\hat{\mathcal{O}}/\mathcal{O})arrow H^{1}P(S^{1}, A_{0})(=\hat{O}/\mathcal{O})arrow 0$

.

101

The dimension is finite and is equal to$= \sup\{i-v(a_{i}) :$i_{0}(P)$i=0, \ldots, m\}-(m-v(a_{m}))$i=0, \ldots, m\}$ ,$(v(a_{m})-m)- \inf\{v(a_{i})-i :$=$which is called the irregularity by Malgrange [17], [18], the invariant of Fuchs by
Levelt [3], [4] or the irregular index
Komatsu (in a private communication), where,$\mathrm{C}_{7\acute{\mathrm{e}}\mathrm{r}\mathrm{a}\Gamma}\mathrm{d}-$$\dot{\mathrm{b}}\mathrm{y}${$v(a)= \sup$ $p:x^{-}a(pX)$isholomorphic at theorigin.}.

Remark : Letand be the ring of the ring of convergent Laurent series withfinite negative order terms, the ring of formal, the ring of formal Laurent series with finite
negative order terms and the ring of convergent Laurent series, respectively. Denote by
one ofand . We consider as an operator from to itself. Then,andare finite dimensional, and has a index;, which can be calculated as follows:$0$$\mathcal{K},\hat{\mathcal{K}}$$\mathcal{E}$$F$$\mathcal{O},\hat{\mathcal{O}},$$\mathcal{K},\hat{\mathcal{K}}$$P$$\mathcal{E}$$F$$\mathrm{K}\mathrm{e}\mathrm{r}(P;F)$$\text{ノ}\chi(P;F)=\mathrm{d}\mathrm{i}\ln c\mathrm{K}\mathrm{e}\mathrm{r}(P$$\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(P;F)$$\dim_{C}\mathrm{C}_{\mathrm{o}\mathrm{k}}\mathrm{e}\mathrm{r}(P;F)$$\chi(P;\mathit{0})$$=$$m-v(a_{m})$ ,$\chi(P;\hat{O})$$=$$\sup\{i-v(a_{i}) :$\chi(P;\mathcal{K})$$=$$m-v(a_{m})- \sup\{i-v(a_{i}) :$\chi(P;\hat{\mathcal{K}})$

$=$

$0$

,

$\chi(P;\mathcal{E})$

$=$

$0$

.

i=1, \ldots, m\}$,i=1, \ldots, ’ n\}$The quantity $i_{0}(P)$ is also equal to the followings [17], $[18]\sim$,$\chi(P;\hat{\mathcal{O}})-x(P;\mathit{0})$$\chi(P;\hat{\mathcal{K}})-\chi(\mathcal{K})$$-\chi(P;\mathcal{K})$,,$\chi(P;\hat{\mathcal{K}}/\mathcal{K})$,$\chi(P;\mathcal{E})-\chi(P;\mathcal{K})$$\chi(P;\mathcal{E}/\mathcal{K})$,,$\chi(P;\mathcal{E}/O)-\chi(P;\mathcal{K}/O)$$\dim_{C}\mathrm{K}\mathrm{e}\mathrm{r}(P;\hat{O}/\mathcal{O})$,$\dim c\mathrm{K}\mathrm{e}\mathrm{r}(P;\hat{\mathcal{K}}/\mathcal{K})$,$\dim c\mathrm{K}\mathrm{e}\mathrm{r}(P;\mathcal{E}/\mathcal{K})$,,$\dim_{C}\mathrm{I}\backslash \nearrow \mathrm{e}\mathrm{i}\backslash (P;(\mathcal{E}/O)/(\mathcal{K}/O))$

.

,

-

102

Remark 1: If we consider a linear ordinary differential operator with coefficients in holomorphic functions at the infinity in the Riemann Sphere and we do not use the variable, the quantity is equal to$t= \frac{1}{x}$$= \sup\{v’(ai)-i:i=0, \ldots, m\}-(v’(a_{m})-m)$$i_{\infty}(P)$$(m-v’(a_{m}))- \inf\{i-v’(ai) :$=$i=0, \ldots, m\}$ ,

where

{

$v’(a)= \sup$ $p:x^{-}a(pX)$holomorphic at theisinfinity.}.

Remark 2: We have also another important quantity associated with the linear ordinary
differential operator $P=(\Sigma_{i=0}^{m}ai(x)(d/dx)^{i})$ . At the origin, we set$k= \sup\{0, \frac{(v(a_{m})-m)-(v(ai)-i)}{m-i} :i=0, \ldots, m-1\}$ ,and at the infinity, we set$k= \sup\{0, \frac{(m-v’(a_{m}))-(i-v/(ai))}{m-i} :i=0, \ldots, \uparrow-1\}$,

which is called the invariant of Katz by G\’erard-Levelt [3], [4] or the order by Sibuya[28], and $k+1$ is called the irregularity by Komatsu [9], [10]. In order to understand theimportance of this quantity, see the above references and also Ramis [24], [25],[11]. Malgrange [21]. In adding a word,$\mathrm{I}<\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{s}\mathrm{u}$$i_{0}(P \rangle\geq k\geq\frac{i_{0}(P)}{m}, mk\geq i_{0}(P)\geq k$.

Letbe the stalk of germs of linear ordinary differential operators with holomorphic
coefficients, and put $\mathcal{M}_{0}=D_{0}/D_{0}P$ . Then,
has a projective resolution$D_{0}$$\mathcal{M}_{0}$$0\vdash \mathcal{M}_{0}\vdash D0arrow^{P}D_{0}arrow 0$

from which, by operating the functor
values inat the origin,$\mathcal{H}om_{D0}’(\cdot,\mathcal{F}_{0})$,, we have the solution complex with$\mathcal{F}$$Sol(\mathcal{M}0, \mathcal{F}0):\mathcal{F}_{0^{arrow^{P}}}\mathcal{F}_{0}arrow 0$.

We have the isomorphism:
Ext 0$($ $\mathcal{M}_{0},\mathcal{F}0$)$\simeq \mathrm{K}\mathrm{e}\mathrm{r}(\mathcal{F}_{0;}P)$,.

$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{M}_{0},\mathcal{F}0)\simeq \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}1^{\cdot}(\mathcal{F}_{0};P)$

Therefore, the index as $D$ -module at the origin,$\chi(\mathcal{M};\mathcal{F})_{0}=\dim_{C}$Exto$(\mathcal{M}0,\mathcal{F}0)-\dim C\mathrm{E}\mathrm{x}\mathrm{t}(\mathrm{l}\mathcal{M}0, F0)$

,

103

is equal to the index $\chi(P;F)$ , and the irregularity as$D$-module at the origin,,$\mathrm{I}\mathrm{r}\mathrm{r}(\mathcal{M})_{0=x}(\mathcal{M}0;\hat{\mathcal{O}})-x(\mathcal{M}_{0};O)$is equal to the irregularity$\mathrm{I}\mathrm{r}\mathrm{r}(P)_{0}$and$\mathrm{I}\mathrm{r}\mathrm{r}(\mathcal{M})0=x(\mathcal{M}0;\hat{\mathcal{K}})-x(\mathcal{M}_{0};\mathcal{K})$$\mathrm{I}\mathrm{r}\mathrm{r}(\mathcal{M})_{0}=\chi(\mathcal{M}0;\mathcal{E})-\chi(\mathcal{M}0;\mathcal{K})$,,$\mathrm{I}\mathrm{r}\mathrm{r}(\mathcal{M})_{0}=x(\mathcal{M}0;\mathcal{E}/\mathcal{O})-\chi(\mathcal{M}_{0};\mathcal{K}/O)$

3.

Indices of holonomic $D$ -modules and their irregularities

Let $D$ be the sheaf of gerlns of linear partial differential operetors with coefficientsbe a holonomic $D$ -module. Theof holomorphic functions on a manifold $M$ and let
modulehas a projective resolution$\mathcal{M}$$\mathcal{M}$$0 \succ \mathcal{M}\succ D^{m_{\mathrm{O}}}\frac{\text{ノ}P_{0}}{\backslash }D^{m_{1}}\frac{\text{ノ}P_{1}}{\backslash }D^{m_{2}}\frac{JP_{2}}{\backslash }$frorn which, by operating the functor,
values in$\mathcal{H}om_{D}(\cdot, \mathcal{F})$:’$\frac{P_{f^{n-1}}}{\backslash }D^{n\tau_{2n}}\succ 0$, we have the solution complex with$\mathcal{F}$$Sol(\mathcal{M}, \mathcal{F})$:$\mathcal{F}^{m_{\mathrm{O}}}\frac{P_{\mathrm{O}_{\mathrm{t}}}^{k}}{\prime}\mathcal{F}^{m_{1}}P^{t}\ldots P_{21}^{t}\underline{1_{\mathrm{t}\prime}}narrow^{-}\mathcal{F}^{m2n}arrow 0$,

For a point , the index of holonomic $D$-module$p$$\mathcal{M}$with values in$\chi(\mathcal{M};\mathcal{F})_{p}=\sum_{i=0}^{2n}\dim C(-1)^{i}\mathcal{E}_{X}t^{i}(\mathcal{M}, \mathcal{F})_{p}$For the point , the$p$$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{g}_{\mathrm{U}}1\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$of holonomic $D$-module$\mathcal{M}$$\mathcal{F}$is defined by.

is defined by$\mathrm{I}\mathrm{r}\mathrm{r}(\mathcal{M})_{P}=\chi(\mathcal{M};\mathit{0}_{M^{\wedge}})|H)p-\chi(\mathcal{M};\mathcal{O}_{M|}H)_{p}$

,

where
is the sheaf of gelms of holomorphic functions on $M,$ $H$ is the set of singularis thepoints ofis the zero-extension of the restriction of $O$ on $H$ andZariski completion of along $H$ .
$\mathcal{O}$$\mathcal{M},$$\mathcal{O}_{M|H}$$\mathcal{O}_{M|H}\wedge$$\mathcal{O}$4

Holonomic $D$ -module defined by Humbert confluent hypergeometric partial differential equations$\Phi_{D}$In the sequel, we consider the solution complexes of holononic
by Humbert confluent hypergeometric partial differential equations) and give the calculation of the cohomology groups.
Lauricella$F_{D}$$D$-module defined(derived from$\Phi_{D}$104

We put $M=(P_{C}^{1})^{m}$ andFor a domain included in$U$, where, we define$H= \bigcup_{k=1}^{m}H_{k}$$H_{k}$$H_{k}=P_{C}^{1}\cross\cdots\cross\{\infty\}\cross\cdots\cross P_{C}^{1}$.

$s.t. \sup_{U\hat{y}k\in}|f_{n}(y_{k})|<CA^{n}\{(n-1)!\}s-1\}$$\mathcal{O}_{\overline{M|H},s,A}(U)=\{\sum_{j\geq 0}f_{j}(y_{k})(x_{k})-j;\exists C>0,\forall n,$,

where $y_{k}=(x_{1}, \cdots, x_{k-}1, Xk+1, \cdots,xm),\hat{y}_{k}=(x_{1}, \cdots,X_{k-}1, \infty, xk+1, \cdots,x_{m})-$ . For a point, if $p\in H_{k}$ then we put$p \in H\backslash \bigcup_{k\neq l}(H_{k}\cap H_{T})$$( \mathcal{O}_{\overline{M|H},s,A})_{p}=\mathrm{I}\mathrm{n}\mathrm{d}\lim_{\subset p\in UHk}\mathcal{O}_{\overline{M|}s}(H,,AU)$.

We define as follow:$(O_{\overline{M|H},s})_{p}$$=$$(o_{\overline{M|H},(s)})p$$=$$(O_{\overline{M|}-}H,S,A)_{p}$$(O_{\overline{M|H},(S,A+)})_{p}$$\mathrm{I}\mathrm{n}\mathrm{d}\lim_{>A0}(O_{\overline{M|}s,A})H,p$$\mathrm{P}\mathrm{r}\mathrm{o}i_{>}^{\lim(}0\mathit{0}_{M})_{p}\overline{|H},s,A$$=’ \mathrm{I}\mathrm{n}\mathrm{d}\lim_{0<B<A}(\mathcal{O})_{p}\overline{M|H},s,B$

$=$

,
’

’

$\mathrm{p}_{\mathrm{r}\mathrm{o},B\dot{1}}\lim(\mathcal{O}_{M}A\overline{|H},s,B)_{\mathrm{P}}$.

The system of Humbert confluent hypergeometric partial differential equations
is as follows:$\Phi_{D}$:$x_{k^{\frac{\partial^{2}u}{\partial x_{k}^{2}}}}+ \sum_{l\neq k}x_{l}\frac{\partial^{2}u}{\partial x_{k}\partial x_{l}}+(c-X_{k})\frac{\partial u}{\partial x_{k}}-b_{k}u=0$(denoted by$\Phi_{D}[1]$$L_{k}.u=0\mathrm{f}\mathrm{o}\mathrm{r}k=1,$$\cdots,$

where $b_{k}(k=1, \cdots, m)$ and are not non-negative integers. Note that
commute$D$with each other. We consider the -module
we putdefined by$c$$L_{k}’ \mathrm{s}$$\Phi_{D},$$\mathcal{M}_{D}$$\mathcal{M}_{D}=D/(DL_{1}+\cdots+DL_{m})$$\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{T}\mathrm{e}\mathrm{l}\mathrm{y}$.

We have a projective resolution like Koszul complex$0 arrow \mathcal{M}_{D}arrow D^{n_{0}}\frac{\text{ノ}\nabla_{0}^{m}}{1}D^{n_{1}}arrow\nabla_{1}^{m}$and we have the solution complex$n_{q}= \frac{m!}{q!(m-q)!}$and$\nabla_{q}^{m}’ \mathrm{s}$$\frac{\nabla^{m}\text{ノ^{}q-1}}{\backslash }D^{n_{q}}arrow\cdotsarrow^{1}-D\nabla_{q}^{m}\nabla^{m}mn_{m}arrow 0$$Sol(\mathcal{M}_{D}, \mathcal{F})$$\mathcal{F}^{n_{0}}arrow \mathcal{F}\nabla_{0}^{m}n_{1}arrow\nabla_{1}^{\mathrm{m}}$

where

...

...

with values in$\nabla_{q-1 ,arrow \mathcal{F}n_{q}}^{m}arrow\nabla_{q}^{m}$...

$\mathcal{F}$$\nabla_{m-1 ,arrow \mathcal{F}^{n_{m}}}^{m}arrow 0$,

are defined by the following recursive manner:$\nabla^{m}=$$\nabla_{00}^{1}=L_{1},$$\cdots,$

$m$

)

105

$\nabla_{1}^{2}=((-1)2-1L_{2}, L_{1}),$ $\cdots,$$\nabla_{1}^{m}=$ .

$\nabla_{q}^{m}=$ .
... ,$\nabla_{m-1}^{m}=((-1)m-1Lm’\nabla m-2m-1)$,

and we have the following
Theorem 1. Let $M=(P_{C}^{1})^{m},$beabove. The
dimensions of chohomology groups of the solution complexes for the $D$ -module defined by
are as folow:(1) If $1\leq s<2$ ,$H= \bigcup_{k=1k,p}^{m}H\in H\backslash \bigcup_{k,l}(H_{k^{\mathrm{s}}}\cap H_{l})$$a\mathit{8}$$\Phi_{D}$for$\mathcal{F}=\mathcal{O}_{\overline{M|H},(S}$)’$\mathcal{O}_{M},$$\mathcal{O}_{\overline{M|}}\overline{|H},s,A-H,(_{S,A+)}$,$O\overline{M|H}_{S},$,1,$(j=0,2, \cdots, m)$$0$$\dim c\mathrm{E}_{\mathrm{X}\mathrm{t}^{j}(}(\mathcal{M}D)p’ \mathcal{F}_{p})=\{$’$(j=1)$(2) If $s>2$ ,

for

$OH,$
)

$\mathcal{F}=\mathcal{O}_{\overline{M|}(s},$$\overline{M|H},s,A-,\overline{M|H},(O, \mathcal{O}_{M}S,A+)\overline{|H},s$$\dim_{C}\mathrm{E}\mathrm{X}\mathrm{t}j((\mathcal{M}_{D})p’ \mathcal{F}_{p})=0$’,$(j=0,1,2,$$m\rangle$.

,$(j=0,1,2, \cdots, m)$.

$\cdots,$(3) In the case of $s=2$ ,(i) if $A>1_{2}$for $\mathcal{F}=O,$$\mathcal{O}\overline{M|H},2,A-\overline{M|H},(2,A+)$’$\dim_{C}\mathrm{E}_{\mathrm{X}\mathrm{t}^{j}(}(\mathcal{M}D)_{p},\mathcal{F}_{p})=0$(ii) if $0<A<1$ ,for$\mathcal{F}=\mathcal{O},$$\mathit{0}\overline{M|H},2,A-\overline{M|H},(2,A+)$

’

,
1,

$0$
$\dim c\mathrm{E}_{\mathrm{X}}\mathrm{t}j((\mathcal{M}D)p’ \mathcal{F}_{p})=\{$$(j=0,2, \cdots, m)$$(j=1)$(iii) if $A=1$ ,

,
1,

$0$

$\dim c\mathrm{E}_{\mathrm{X}}\mathrm{t}j((\mathcal{M}D)_{p}, (O_{\overline{M|}-})_{p})H,2,1=\{$$(j=0,2, \cdots, m)$$(j=1)$

106$\dim_{C}\mathrm{E}_{\mathrm{X}}\mathrm{t}^{j}((\mathcal{M}D)_{p}, (O_{\overline{M|H},(2,1})_{p})+)=0$$((\mathcal{M}_{D})p’(\mathcal{O}_{\overline{M|H},(2})_{P}))=\{$$\dim_{C}\mathrm{E}\mathrm{x}\mathrm{t}^{j}((\mathcal{M}_{D})_{p}, (O_{\overline{M|H},2})_{p})=0$

(4)

,

,

$(j=0,2, \cdots, m)$1,$(j=1)$$(j=0,1,2, \cdots, m)$$(j=0,1,2, \cdots, m)$$\dim_{C}\mathrm{E}_{\mathrm{X}}\mathrm{t}^{j}((\mathcal{M}D)_{p}, (\mathcal{O}_{\overline{M|H}})_{p})=0$

Corollary 1. The indexes of $D$ -module defined by(1) If $1\leq s<2$ ,for$(j=0,1,2, \cdots, \uparrow\eta)$,$0$(iv) $\dim c$ Ext,$\Phi_{2}$.

are as follow:$o_{M}H,\overline{M|H}_{S,A}-\overline{|H},(_{S,A}+)\overline{|H},s$$\mathit{0},’ \mathcal{O}_{M},$’$\mathcal{F}=\mathit{0}_{\overline{M|}(S)},$

X$((\mathcal{M}_{D})_{p}, \mathcal{F}_{p})=-1$.

(2) If $s>2$ ,for$\mathcal{F}=\mathit{0},$$\mathit{0}_{\overline{M|}}\overline{M|H},(_{S})H,s,A-,\overline{M|H},(o, os,A+)\overline{\Lambda f|H},S$$\mathcal{X}((\mathcal{M}_{D})_{p}, \mathcal{F}_{p})=0$’.

(3) In the case of $s=2$(i) if $A>1$ ,for$\mathcal{F}=\mathcal{O},$$\mathit{0}\overline{M|H},2,A-\overline{M|H},(2,A+)$’$\mathcal{X}((\mathcal{M}_{D})_{pp}, \mathcal{F})=0$.

(ii) if $0<A<1$ ,for$\mathcal{F}=O_{\overline{M|H}},’ O2,A-\overline{M|H},(2,A+)$ ’$\mathcal{X}((\mathcal{M}_{D})_{p}, \mathcal{F}_{p})=-1$.

(iii) if $A=1$ ,X $((\mathcal{M}_{D})_{p}, (o)\overline{M|H},2,1-p)=-1$ .

X

(iv)

$((\mathcal{M}_{D})_{p}, (\mathcal{O}_{\overline{M|}()})H,2,1+p)=0$$\mathcal{X}((\mathcal{M}_{D})_{P}, (\mathcal{O})_{p}\overline{M\mathrm{I}H},(2))=-1$$\mathcal{X}((\mathcal{M}_{D})p’(O_{\overline{M|H},2})p)=0$(4)$\mathcal{X}((\mathcal{M}_{D})p’(o\overline{M|H})_{\mathrm{p}})=0$

Corollary 2

.

.

.

The irregularity$\mathrm{I}\mathrm{r}\mathrm{r}((\mathcal{M}_{D})_{p})=1$

.

.

.

.

107

as follow:
The essential parts for1. We have a formal solution$k=\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{e}$\^u $(x)= \sum_{n}\infty=1\frac{(n-1)!(_{C}-n)\cdots(c-2)}{(n-b_{1})\cdots(1-b1)}\Phi mD^{-1}(b_{2}, \cdots, b_{m}; c-n;X2, \cdots, X_{m})(_{X_{1}})^{-}n$of the non-homogeneous system of partial differential equations$L_{1}$\^u $(x)= \frac{\Phi_{D}^{m-1}(b_{2},\cdots,b\cdot c-m}{x_{1}}$ ,$L_{l}\hat{u}(x)=0(l=2, \cdots, m)$,where $\Phi_{D}^{m-1}(b_{2}, \cdots, b_{m};c-1;x_{2}, \cdots, X_{m})$ is the Humbert confluent hypergeoetric functionin $(m-1)$ variables with the parameter $(b_{2}, \cdots, b_{m};c-1)$ ,$\Phi_{D}^{m-1}(b_{2}, \cdots, b_{m};C-1;x_{2}, \cdots, X_{m})=\sum_{j2=0}\infty$

. ..

,

$j_{m}= \sum_{0}^{\infty}\frac{(b_{2})_{j_{2}}\cdots(b)_{j_{m}}\eta(x_{2})j2\ldots(xm)^{j}m}{(c)_{j\mathrm{o}+}\sim\ldots+j_{m}j2!\cdots j_{n?}!}$

where $(b)_{s}=(b+1)\cdots(b+s-1)$ .
2. If, for$v=$$\nabla_{1}^{m}v=0$, we have$\nabla_{0}^{m}(\sum_{j}\infty f=0j(y1)x_{1}-j)=v$, then$\frac{(1-b_{1})\cdots(n.+.1-b_{1})}{n!(c-n-1)\cdot(c-2)}f_{n+1}$.

$= \sum_{j=1}^{n+}1\frac{(1-b_{1})\cdots(j-.1.-b_{1})}{(j-1)!(C--j)\cdot(_{C}-2)}P_{j}^{1}+\sum_{l=2}^{m}X_{l^{\frac{\partial}{\partial x_{l}}}}j1\sum_{=}^{n}\frac{(1-b_{1})\cdots(j.-b1)}{(j-1)!(_{C}-j)\cdot\cdot(_{C}-2)}f_{j}$Put$F_{n+1}= \frac{(1-b_{1})\cdots(n+1-b_{1})}{n!(\mathrm{c}-n-1)\cdots(C-2)}f_{n+1}$, then, for $l=2,$$\cdot\cdot,$
$,$

$m$

,

$\frac{\partial}{\partial x_{l}}F_{n+1}=\frac{1}{c-n-1}(x_{\iota^{\frac{\partial}{\partial x_{l}}}}p_{n}+b_{l}F_{n}+P_{n}^{l}+x_{l^{\frac{\partial}{\partial x_{l}}}}P_{n+1}^{1})$andis well-defined as constant for
and, $0<s<2$ or ( $s=2$ and $0<A<1$ ).
$\alpha(v)=\lim_{narrow\infty}F_{n}$,$v\in(\mathcal{O}_{\overline{M|H},s},(AU))^{m}$, where$U\subset H_{1}$$=’\not\equiv_{\overline{\overline{\mathrm{X}}}}\wedge\ovalbox{\tt\small REJECT}’\vee^{\backslash }$[1] Appell,

P-E., and Kamp\’e de F\’eriet, J.: Fonctions hyperg\’eom\’etriques et hypersph\’eriques, Gauthier-Villars , (1926).

[2] Deligne, P.: Equations diff\’erentielles \‘a points singuliers r\’eguliers, Lecture Notes in
Math., no. 163, Springer-Verlag, (1970).

108[3] Ge’rard, R. and Levelt, A.: Mesure de l’irregularit\’e en un point singulier d’un
. Acad. Sc. Paris, t. 274 (1972),
syste\‘eme d’equations diff\’erentielles line’aires,pp. 774-776, pp. 1170-1172.
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[6] Ishizuka, S. and Majima, H. , On the solution complexes of confluent hypergeometricof Holomorphic Vector Fields and-modules, RIMS Kokyuroku
Related Topics, editedd by T. Suwa) 1994, pp10-19$D$$878(\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}$[7] Kashiwara, M., Algebraic study for systems of partial differential equations, Master’s
thesis University of Tokyo, 1971.
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$\mathrm{I}\mathrm{A}$[9] Komatsu, H.: An introduction to the theory of hyperfunctions, Hyperfunctions and, 1971,
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$\mathrm{I}<\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{a}$[10] Komatsu, H.: On the regularity of hyperfunction solutions of linear ordinary differential equations with real analytic coefficients, J. Fac. Sci. Univ. Tokyo, Sect. ,Vol. 20, (1973), pp.107-119.
$\mathrm{I}\mathrm{A}$[11] Komatsu, H.: Linear Ordinay Differential Equations with Gevrey Coefficients, J. Diff.
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[12] Majima, H.: Vanishing theorems in asymptotic analysis, Proc. Japan Acad., 59 Ser.
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109[16] Majima. H., Howls, C. J. and Olde Daalhuis, A. B.: Vanishing Theorem in Asymptotic Analysis III (to appear in “Structure of Solutions of Differential Equations”
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$\mathrm{C}.\mathrm{R}$.

[18] Malgrange, B.: Sur les points singuliers des \’equations diff\’erentielles, l’Enseignment
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[19] Malgrange, B.: Remarques sur les Equations Diff\’erentielles \‘a Points Singuliers, in Equations Diff\’erentielles et Syst\‘emes de Pfaff daloe le Champ Complexe
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