Irregularities on hyperlanes of holonomic $\mathcal{D}$-module : especially defined by confluent hypergeometric partial differential equations (Complex Analysis and Microlocal Analysis)

(1)

Irregularities

on

hyperlanes

of

holonomic

$D$

-module

(especially

defined

by

confluent

hypergeometric

partial

differential

equations)-お茶の水女子大学理学部数学科

真島

秀行

(Hideyuki Majima,

Ochanomizu

University)

1 Intoduction

In this expositorypaper, I will explain the irregularity at asingular point ofdifferential

equation. At first, I will give you a review of study on ordinary linear differential

equa-tions. Secondly, I will talk about holonomic $D$-modules, especially, Humbert confluent

hypergeometric differential modules in $m$ variables.

2 Irregularity

of

holonomic

$D$

-module defined by

an

ordinary

differential operator.

Consider a linear ordinary differential operator with coefficients in holomorphic func-tions at the origin in the Riemann Sphere:

$Pu=( \sum_{i=0}^{m}a_{i}(X)(d/dx)i)u$.

where $a_{m}$ is supposed not to be identically zero. Let $O$ and

$\hat{O}$ be the ring of convergent

power-series and the ring of formal power-series in $x$, respectively. Then, we see the

following isomorphism oflinear spaces due to Deligne (cf. [23], etc.) :

$H^{1}(S^{1}, \mathcal{K}er(P:A\mathrm{o}))\simeq \mathrm{K}\mathrm{e}\mathrm{r}(P;\hat{\mathcal{O}}/O)$ ,

for, from the existence theorem of asymptotic solutions due to Hukuhara (cf. [26]) (and

other many contributers), we have the short exact sequence

$\mathrm{O}arrow \mathcal{K}er(P:A\mathrm{o})arrow A_{0}-^{P}A_{0}arrow 0$, fromwhich,

we

get the exact sequence,

(2)

The dimension is finite and is equal to

$i_{0}(P)$ $= \sup\{i-v(a_{i}) : i=0, \ldots, m\}-(m-v(a_{m}))$

$=$ $(v(a_{m})-m)- \inf\{v(a_{i})-i : i=0, \ldots, m\}$,

which is called the irregularity by Malgrange [17], [18], the invariant of Fuchsby $\mathrm{C}_{7\acute{\mathrm{e}}\mathrm{r}\mathrm{a}\Gamma}\mathrm{d}-$

Levelt [3], [4] or the irregular index $\dot{\mathrm{b}}\mathrm{y}$

Komatsu (in a private communication), where,

$v(a)= \sup$

{

$p:x^{-}a(pX)$ is holomorphic at the

origin.}.

Remark $0$: Let $\mathcal{K},\hat{\mathcal{K}}$ and $\mathcal{E}$ be the ring of the ring of convergent Laurent series with

finitenegative order terms, the ringof formal, thering offormal Laurent serieswith finite

negative order terms and the ring ofconvergent Laurent series, respectively. Denote by $F$

one of$\mathcal{O},\hat{\mathcal{O}},$ $\mathcal{K},\hat{\mathcal{K}}$ and $\mathcal{E}$

.

Weconsider $P$as anoperator from$F$to itself. Then,

$\mathrm{K}\mathrm{e}\mathrm{r}(P;F)$

and $\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(P;F)$ are finite dimensional, and has a index $\text{ノ}\chi(P;F)=\mathrm{d}\mathrm{i}\ln c\mathrm{K}\mathrm{e}\mathrm{r}(P$;

-$\dim_{C}\mathrm{C}_{\mathrm{o}\mathrm{k}}\mathrm{e}\mathrm{r}(P;F)$

,

which

can

be calculated as follows:

$\chi(P;\mathit{0})$ $=$ $m-v(a_{m})$,

$\chi(P;\hat{O})$ $=$ $\sup\{i-v(a_{i}) : i=1, \ldots, m\}$,

$\chi(P;\mathcal{K})$ $=$ $m-v(a_{m})- \sup\{i-v(a_{i}) : i=1, \ldots, ’ n\}$, $\chi(P;\hat{\mathcal{K}})$ _$=$ $0$,

$\chi(P;\mathcal{E})$ $=$ $0$.

The quantity $i_{0}(P)$ is also equal to the followings [17], $[18]\sim$

$\chi(P;\hat{\mathcal{O}})-x(P;\mathit{0})$, $\chi(P;\hat{\mathcal{K}})-\chi(\mathcal{K})$, $-\chi(P;\mathcal{K})$, $\chi(P;\hat{\mathcal{K}}/\mathcal{K})$, $\chi(P;\mathcal{E})-\chi(P;\mathcal{K})$, $\chi(P;\mathcal{E}/\mathcal{K})$, $\chi(P;\mathcal{E}/O)-\chi(P;\mathcal{K}/O)$, $\dim_{C}\mathrm{K}\mathrm{e}\mathrm{r}(P;\hat{O}/\mathcal{O})$, $\dim c\mathrm{K}\mathrm{e}\mathrm{r}(P;\hat{\mathcal{K}}/\mathcal{K})$ , $\dim c\mathrm{K}\mathrm{e}\mathrm{r}(P;\mathcal{E}/\mathcal{K})$

,

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Remark 1: If we consider a linear ordinary differential operator with coefficients in

holo-morphic functions at the infinity in the Riemann Sphere and we do not use the variable

$t= \frac{1}{x}$

,

the quantity is equal to

$i_{\infty}(P)$ $= \sup\{v’(ai)-i:i=0, \ldots, m\}-(v’(a_{m})-m)$

$=$ $(m-v’(a_{m}))- \inf\{i-v’(ai) : i=0, \ldots, m\}$,

where

$v’(a)= \sup$

{

$p:x^{-}a(pX)$ is holomorphic at the

infinity.}.

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 the

importance of this quantity, see the above references and also Ramis [24], [25], $\mathrm{I}<\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{s}\mathrm{u}$

[11]. Malgrange [21]. In adding a word,

$i_{0}(P \rangle\geq k\geq\frac{i_{0}(P)}{m}, mk\geq i_{0}(P)\geq k$

.

Let $D_{0}$ bethe stalk ofgerms oflinear ordinarydifferential operators with holomorphic

coefficients, and put $\mathcal{M}_{0}=D_{0}/D_{0}P$

.

Then, $\mathcal{M}_{0}$ has aprojective resolution

$0\vdash \mathcal{M}_{0}\vdash D0arrow^{P}D_{0}arrow 0$,

from which, by operating the functor $\mathcal{H}om_{D0}’(\cdot,\mathcal{F}_{0})$, we have the solution complex with

values in $\mathcal{F}$ at the origin,

$Sol(\mathcal{M}0, \mathcal{F}0):\mathcal{F}_{0^{arrow^{P}}}\mathcal{F}_{0}arrow 0$

.

We have the isomorphism:

Ext0$($$\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,

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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

irregu-larities

Let $D$ be the sheaf of _gerlns of linear partial differential operetors with coefficients

of holomorphic functions on a manifold $M$ and let $\mathcal{M}$ be a holonomic $D$-module. The

module $\mathcal{M}$ has a projective resolution

$0 \succ \mathcal{M}\succ D^{m_{\mathrm{O}}}\frac{\text{ノ}P_{0}}{\backslash }D^{m_{1}}\frac{\text{ノ}P_{1}}{\backslash }D^{m_{2}}\frac{JP_{2}}{\backslash }$ :’$\frac{P_{f^{n-1}}}{\backslash }D^{n\tau_{2n}}\succ 0$

frorn which, by operating the functor $\mathcal{H}om_{D}(\cdot, \mathcal{F})$

,

we have the solution complex with

values in $\mathcal{F}$ ,

$Sol(\mathcal{M}, \mathcal{F})$

:

$\mathcal{F}^{m_{\mathrm{O}}}\frac{P_{\mathrm{O}_{\mathrm{t}}}^{k}}{\prime}\mathcal{F}^{m_{1}}P^{t}\ldots P_{21}^{t}\underline{1_{\mathrm{t}\prime}}narrow^{-}\mathcal{F}^{m2n}arrow 0$

,

For a point$p$, the index of holonomic $D$-module $\mathcal{M}$with values in $\mathcal{F}$ is defined by

$\chi(\mathcal{M};\mathcal{F})_{p}=\sum_{i=0}^{2n}\dim C(-1)^{i}\mathcal{E}_{X}t^{i}(\mathcal{M}, \mathcal{F})_{p}$

.

For the point $p$, the $\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{g}_{\mathrm{U}}1\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$ of holonomic $D$-module $\mathcal{M}$ 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 $\mathcal{O}$ is the sheaf of

gelms of holomorphic functions on $M,$ $H$ is the set of singular

points of $\mathcal{M},$ $\mathcal{O}_{M|H}$ is the zero-extension of the restriction of $O$

on

$H$ and $\mathcal{O}_{M|H}\wedge$ is the

Zariski completion of$\mathcal{O}$ along $H$

.

4 Holonomic

$D$

-module

defined

by

Humbert

conflu-ent

hypergeometric partial differential equations

$\Phi_{D}$

In the sequel, we consider the solution complexes of holononic $D$-module defined

by Humbert confluent hypergeometric partial differential equations $\Phi_{D}$ (derived from

(5)

We put $M=(P_{C}^{1})^{m}$ and $H= \bigcup_{k=1}^{m}H_{k}$, where $H_{k}=P_{C}^{1}\cross\cdots\cross\{\infty\}\cross\cdots\cross P_{C}^{1}$.

For a domain $U$ included in $H_{k}$, we define

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

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

$p \in H\backslash \bigcup_{k\neq l}(H_{k}\cap H_{T})$

,

if$p\in H_{k}$ then we put

$( \mathcal{O}_{\overline{M|H},s,A})_{p}=\mathrm{I}\mathrm{n}\mathrm{d}\lim_{\subset p\in UHk}\mathcal{O}_{\overline{M|}s}(H,,AU)$

.

We define as follow: $(O_{\overline{M|H},s})_{p}$ $=$ $\mathrm{I}\mathrm{n}\mathrm{d}\lim_{>A0}(O_{\overline{M|}s,A})H,p$ , $(o_{\overline{M|H},(s)})p$ $=$ $\mathrm{P}\mathrm{r}\mathrm{o}i_{>}^{\lim(}0\mathit{0}_{M})_{p}\overline{|H},s,A$ _’ $(O_{\overline{M|}-}H,S,A)_{p}$ $=’ \mathrm{I}\mathrm{n}\mathrm{d}\lim_{0<B<A}(\mathcal{O})_{p}\overline{M|H},s,B$ ’ $(O_{\overline{M|H},(S,A+)})_{p}$ $=$ $\mathrm{p}_{\mathrm{r}\mathrm{o},B\dot{1}}\lim(\mathcal{O}_{M}A\overline{|H},s,B)_{\mathrm{P}}$.

The system of Humbert confluent hypergeometric partial differential equations $\Phi_{D}[1]$

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$L_{k}.u=0\mathrm{f}\mathrm{o}\mathrm{r}k=1,$

$\cdots,$$m$)

where $b_{k}(k=1, \cdots, m)$ and $c$ are not non-negative integers. Note that $L_{k}’ \mathrm{s}$ commute

with each other. We consider the $D$-module $\mathcal{M}_{D}$ defined by $\Phi_{D},$ $\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{T}\mathrm{e}\mathrm{l}\mathrm{y}$we put

$\mathcal{M}_{D}=D/(DL_{1}+\cdots+DL_{m})$.

We have a projective resolution like Koszul complex

$0 arrow \mathcal{M}_{D}arrow D^{n_{0}}\frac{\text{ノ}\nabla_{0}^{m}}{1}D^{n_{1}}arrow\nabla_{1}^{m}$

...

$\frac{\nabla^{m}\text{ノ^{}q-1}}{\backslash }D^{n_{q}}arrow\cdotsarrow^{1}-D\nabla_{q}^{m}\nabla^{m}mn_{m}arrow 0$

and we have the solution complex $Sol(\mathcal{M}_{D}, \mathcal{F})$ with values in $\mathcal{F}$ $\mathcal{F}^{n_{0}}arrow \mathcal{F}\nabla_{0}^{m}n_{1}arrow\nabla_{1}^{\mathrm{m}}$

...

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

...

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

,

where $n_{q}= \frac{m!}{q!(m-q)!}$ and $\nabla_{q}^{m}’ \mathrm{s}$ are defined by the following recursive manner:

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

$\nabla_{1}^{m}=$

.

$\nabla_{q}^{m}=$

.

...

,

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

and we have the following

Theorem

1.

Let $M=(P_{C}^{1})^{m},$ $H= \bigcup_{k=1k,p}^{m}H\in H\backslash \bigcup_{k,l}(H_{k^{\mathrm{s}}}\cap H_{l})$ be $a\mathit{8}$

above.

The

dimensions

_of

chohomologygroups

_of

the solution complexes

_for

the $D$-module

defined

by

$\Phi_{D}$

are

as

folow:

(1)

_If

$1\leq s<2$,

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},$_’

$\dim c\mathrm{E}_{\mathrm{X}\mathrm{t}^{j}(}(\mathcal{M}D)p’ \mathcal{F}_{p})=\{$

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

1, $(j=1)$

(2)

_If

$s>2$,

for

$\mathcal{F}=\mathcal{O}_{\overline{M|}(s},$$OH,$

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

$\cdots,$$m\rangle$.

(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+)$_’

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

.

(ii)

_if

$0<A<1$

,

for

$\mathcal{F}=\mathcal{O},$$\mathit{0}\overline{M|H},2,A-\overline{M|H},(2,A+)$_’ $\dim c\mathrm{E}_{\mathrm{X}}\mathrm{t}j((\mathcal{M}D)p’ \mathcal{F}_{p})=\{$

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

1, $(j=1)$

(iii)

_if

$A=1$,

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

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

(7)

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

.

(iv) $\dim c$Ext $((\mathcal{M}_{D})p’(\mathcal{O}_{\overline{M|H},(2})_{P}))=\{$

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

1, $(j=1)$

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

.

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

.

Corollary 1. The indexes

_of

$D$-module

defined

by $\Phi_{2}$ are as

follow:

(1)

_If

$1\leq s<2$,

for

$\mathcal{F}=\mathit{0}_{\overline{M|}(S)},$$o_{M}H,\overline{M|H}_{S,A}-\overline{|H},(_{S,A}+)\overline{|H},s$ ’$\mathit{0},’ \mathcal{O}_{M},$ 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$((\mathcal{M}_{D})_{p}, (\mathcal{O}_{\overline{M|}()})H,2,1+p)=0$.

(iv) $\mathcal{X}((\mathcal{M}_{D})_{P}, (\mathcal{O})_{p}\overline{M\mathrm{I}H},(2))=-1$

.

$\mathcal{X}((\mathcal{M}_{D})p’(O_{\overline{M|H},2})p)=0$

.

(4) $\mathcal{X}((\mathcal{M}_{D})p’(o\overline{M|H})_{\mathrm{p}})=0$.

(8)

The essential parts for $k=\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{e}$ as follow:

1. We have a formal solution

\^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 function in $(m-1)$ 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$,

$\frac{\partial}{\partial x_{l}}F_{n+1}=\frac{1}{c-n-1}(x_{\iota^{\frac{\partial}{\partial x_{l}}}}p_{n}+b_{l}F_{n}+P_{n}^{l}+x_{l^{\frac{\partial}{\partial x_{l}}}}P_{n+1}^{1})$ ,

and $\alpha(v)=\lim_{narrow\infty}F_{n}$ is well-defined

as

constant for _{$v\in(\mathcal{O}_{\overline{M|H},s},(AU))^{m}$}, where $U\subset H_{1}$

and,

$0<s<2$

or ($s=2$and

$0<A<1$

).

$=’\not\equiv_{\overline{\overline{\mathrm{X}}}}\wedge\ovalbox{\tt\small REJECT}’\vee^{\backslash }$

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