Twisted Euler transform of differential equations with an irregular singular point
aa r X i v : . [ m a t h . C A ] D ec Twisted Euler transform of differential equationswith an irregular singular point
Kazuki Hiroe ∗ Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1Komaba, Meguro-ku, Tokyo, 153-8914, Japan.
Abstract
In [8], N. Katz introduced the notion of the middle convolution onlocal systems. This can be seen as a generalization of the Euler trans-form of Fuchsian differential equations. In this paper, we consider thegeneralization of the Euler transform, the twisted Euler transform, andapply this to differential equations with irregular singular points. Inparticular, for differential equations with an irregular singular pointof irregular rank 2 at x = ∞ , we describe explicitly changes of localdatum caused by twisted Euler transforms. Also we attach these dif-ferential equations to Kac-Moody Lie algebras and show that twistedEuler transforms correspond to the actions of Weyl groups of these Liealgebras. For a function f ( x ), the following integral I λa f ( x ) = 1Γ( λ ) Z xa ( x − t ) λ − f ( t ) dt is called the Riemann-Liouville integral for a, λ ∈ C . If we take a function f ( x ) = ( x − a ) α φ ( x ) where α ∈ C \ Z < and φ ( x ) is a holomorphic functionon a neighborhood of x = a and φ ( a ) = 0, then it is known that I − na f ( x ) = d n dx n f ( x ) . Hence one can consider the Riemann-Liouville integral to be a fractional orcomplex powers of derivation ∂ = ddx . This may allow us to write ∂ λ f ( x ) = I − λa f ( x ) formally. ∗ E-mail: [email protected] ∂ λ p ( x ) ψ ( x ) = n X i =0 (cid:18) λi (cid:19) p ( i ) ( x ) ∂ λ − i φ ( x ) , if p ( x ) is a polynomial of degree equal to or less than n .Now let us consider a differential operator with polynomial coefficients, P ( x, ∂ ) = n X i =0 a i ( x ) ∂ i . The above Leibniz rule assures that ∂ λ + m P ( x, ∂ ) ∂ − λ gives the new differential operator with polynomial coefficients if we choosea suitable m ∈ Z . Moreover if f ( x ) satisfies P ( x, ∂ ) f ( x ) = 0 and I − λa f ( x ) iswell-defined for some a, λ ∈ C , then we can see that ∂ λ + m P ( x, ∂ ) ∂ − λ I − λa f ( x ) = ∂ λ + m P ( x, ∂ ) ∂ − λ + λ f ( x )= ∂ − λ + m P ( x, ∂ ) f ( x )= 0 . Hence ∂ λ turns a differential equation with polynomial coefficients P ( x, ∂ ) u =0 into a new differential equation with polynomial coefficients Q ( x, ∂ ) u = 0,and moreover a solution of Q ( x, ∂ ) u = 0 is given by a solution of P ( x, ∂ ) u =0 if the Riemann-Liouville integral is well-defined. This correspondence ofdifferential equations is called the Euler transform.For example, let us take the differential equation of the Gauss hyperge-ometric function, x (1 − x ) ∂ u + ( γ − ( α + β + 1) x ) ∂u − αβu = 0 . (1.1)Then we can see that ∂ − β ( x (1 − x ) ∂ + ( γ − ( α + β + 1) x ) ∂ − αβ ) ∂ β − = x (1 − x ) ∂ + (( γ − β ) − ( α − β + 1) x ) . And it is not hard to see that the general solution of x (1 − x ) ∂ + (( γ − β ) − ( α − β + 1) x ) u = 0 is given by constant multiples of x β − γ (1 − x ) α − γ +1 .Hence solutions of (1.1) are I β − c x β − γ (1 − x ) α − γ = 1Γ( − β ) Z xc t β − γ (1 − t ) α − γ ( x − t ) − β dt for c = 0 , , ∞ . 2his argument tells us that by the Euler transform ∂ β − , we can reduce(1.1) to an “easier” one, x (1 − x ) ∂ + (( γ − β ) − ( α − β + 1) x ) . And solutions of (1.1) can be obtained from this “easy” equation.This can be applicable to differential equations with an irregular singularpoint. For example, let us consider the differential equations, x∂ u + ( γ − x ) ∂u − αu = 0 , (1.2) ∂ u − x∂u + αu = 0 . (1.3)The first one is the differential equation of the Kummer confluence hyperge-ometric function. The second one is the one of the Hermite-Weber function.Then we have ∂ − α ( x∂ + ( γ − x ) ∂ − α ) ∂ α − = x∂ + (( γ − α ) − x ) , and ∂ α ( ∂ − x∂ + α ) ∂ − α − = ∂ − x. Solutions of these differential equations are x α − γ e x and e x up to constantmultiples. Hence solutions of (1.2) and (1.3) are given by1Γ( − α ) Z xc ( x − t ) − α t α − γ e t dt for c = 0 , ∞ , and1Γ( α ) Z x −∞ ( x − t ) α e t dt = 1Γ( α ) Z ∞ t α e ( x − t )22 dt = e x Γ( α ) Z ∞ e t − tx t α dt. These facts tell us that the Euler transform is a good tool to study asolution of a differential equation from a solution of an easier differentialequation. Then a question arises.Can one always reduce differential equations to “easy” ones by Eulertransforms? If not, find a good class of differential equations which can bereduced to easy one. 3n answer of this question is known for Fuchsian differential equations.Let us consider a system of Fuchsian differential equations of the form, ddx Y ( x ) = r X i =1 A i ( x − c i ) Y ( x ) , (1.4)where A i are n × n complex matrices, Y ( x ) is a C n -valued function and c i ∈ C . In this case, it is known that the Euler transform corresponds to theadditive middle convolution (see [5] and [6]). The answer of our question isgiven by the following theorem. Theorem 1.1 (Katz [8], Dettweiler-Reiter [5] [6]) . The tuple of n × n ofmatrices, A , . . . , A r is irreducible and linearly rigid, the differential equation ( ) can be reduced to ddx y ( x ) = r X i =1 a i x − c i y ( x ) where a i ∈ C by finite iterations of middle convolutions and additions. Definitions of terminologies in this theorem can be found in the originalpapers [5] and [6].Our purpose of this paper is to extend this theorem to non-Fuchiandifferential equations.In this paper, we consider differential equations with polynomial coeffi-cients which have an irregular singular point of rank at most 2 at x = ∞ andsome regular singular points in C . And we give a generalization of Theorem1.1.The organization of this paper is the following. In Section 2, we givea review of some operations on the Weyl algebra. These operations areprovided by T. Oshima in [11] to define the Euler transform in the strictway as an operation on the Weyl algebra. We use these operations to definea generalization of the Euler transform, the twisted Euler transform.Because we treat differential equations with an irregular singular point,solutions of them have asymptotic expansions as formal power series. Hencein Section 3, we study the formal solutions of differential equations. Weintroduce the notion of semi-simple characteristic exponents here.Section 4 is one of the main parts of this paper. We focus on differentialequations which have an irregular singular point of rank 2 at infinity andregular singular points in C . We also assume that at the irregular singularpoint, every formal solutions are “normal type”, i.e., formal solutions can bewritten by e p ( x ) x − µ P ∞ s =0 c s x − s for polynomials p ( x ). Equivalently to say,we assume that differential equations are unramified according to the ter-minology of the differential Galois theory. Then in Section 4, we investigatethe way how the twisted Euler transform changes differential equations, in4ther words, how characteristic exponents at singular points are changed bythe twisted Euler transform.In Section 5, we see the relation between differential equations and Kac-Moody Lie algebras. In [4], W. Crawley-Boevey find a correspondence ofthe systems of Fuchsian differential equations as (1.4) and representationsof quivers and moreover Kac-Moody Lie algebras associated with these rep-resentations. And he solved the existence problem of systems of differentialequations, so-called Deligne-Simpson problem by using the theory of repre-sentations of quivers. An analogous work is done by P. Boalch in [3]. Boalchgeneralizes the result of Crawley-Boevey for the cases which allow an irreg-ular singular point of rank 2. And he also solved Deligne-Simpson problemwith an irregular singular point of rank 2.As an analogue of their works, we attach a Kac-Moody Lie algebra tothe concerning differential equation. Moreover we show that twisted Eulertransforms correspond to simple reflections on the Cartan subalgebra of thisLie algebra. By using this correspondence, we prove the following maintheorem. Theorem 1.2.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12. If idx P > , then P ( x, ∂ ) can be reduced to ( ∂ − αx − β ) n for some α, β ∈ C and n ∈ Z > by finite iterations of twisted Euler trans-forms and additions at regular singular points. Notations used in this theorem are explained in the subsequent sections.In Section 6, we consider the confluence of Fuchsian differential equa-tions. It is known that differential equations of Kummer confluent hypergeo-metric function and Hermite weber functions are obtained from the differen-tial equation of the Gauss hypergeometric function by the limit transitions.In Section 6, as a generalization of this fact, we show the following.
Theorem 1.3.
Take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12. If idx P > ,then P ( x, ∂ ) can be obtained by the limit transition of a Fuchsian Q ( x, ∂ ) ∈ W [ x, ξ ] of idx Q = idx P. Meanwhile, in Appendix, we consider the differential equation which hasregular singular point at x = ∞ and arbitrary singularities at any otherpoints. And we give a necessary and sufficient condition to reduce the orderof this differential equation by Euler transform E (0 , µ ). Theorem 1.4.
Let us take P ( x, ∂ ) ∈ W [ x ] which has regular singular pointat x = ∞ and semi-simple exponents { [ µ ] n , . . . , [ µ l ] n l } , here P li =1 n i = n = ord P , µ i / ∈ Z and µ i − µ j / ∈ Z if i = j . Then we have ord E (0 , µ i − P ( x, ∂ ) < ord P if and only if deg P − ord P < n i . There are many other works about middle convolutions, equivalently tosay, Euler transforms of differential equations with irregular singular points.T. Kawakami considers generalization of middle convolutions to systems ofdifferential equations with irregular singular point which are called general-ized Okubo systems in [9]. K. Takemura [13] and D. Yamakawa [15] considerthe middle convolutions for the system of the form ddx Y ( x ) = r X i =1 k i X j =1 A ij ( x − c i ) j Y ( x )where A ij are n × n complex matrices. In particular, Yamakawa discussesthe reduction of the rank of a differential equation which is similar problemto ours. Acknowledgement
The author is very grateful to Professor Toshio Oshima who kindly taughtthe author his theory of Euler transforms of Fuchsian differential equations.The author also thanks Shinya Ishizaki. The author studied the theory ofmiddle convolutions and Euler transforms on his seminar under the directionof Professor Oshima. Finally the author thanks Noriyuki Abe and RyosukeKodera for the introduction of the Kac-Moody theory.
In this section, some operations on localized Weyl algebra are introduced.In [11], T. Oshima uses these operators to understand the Euler transformand he constructs a theory of the Euler transform which corresponds tothe theory of additive middle convolutions for Schleginger type systems ofdifferential equations studied by Dettweiler and Reiter in [5] and [6].The Weyl algebra W [ x ] is the C -algebra generated by x and ∂ = ddx withthe relation, [ ∂, x ] = ∂x − x∂ = 1 . The localization of the Weyl algebra is W ( x ) = C ( x ) ⊗ C W [ x ] where C ( x )is the quotient field of C [ x ] the polynomial ring with complex coefficients.For indeterminants ξ = ( ξ , . . . , ξ n ), we consider C ( ξ ) the field of rational6unctions with complex coefficients and fix an algebraic closure C ( ξ ) of C ( ξ ).Then we can define the Weyl algebra with parameters W [ x, ξ ] = C ( ξ ) ⊗ C W [ x ] . We can also define the localized Weyl algebra with parameters W ( x, ξ ) = C ( ξ ) ⊗ C W ( x ).The element P ∈ W ( x, ξ ) (resp. P ∈ W [ x, ξ ]) is uniquely written by P = m X i =0 p i ( x ; ξ ) ∂ i with p i ( x, ξ ) ∈ C ( x, ξ ) = C ( ξ ) ⊗ C C ( x ) (resp. p i ( x, ξ ) ∈ C [ x, ξ ] = C ( ξ ) ⊗ C C [ x ]) for i = 0 , . . . , m . According to this representation, the order of P denoted by ord P is defined by the maximum integer i such that p i ( x, ξ ) = 0.We also define the degree of P ∈ W [ x ; ξ ] by maximum of degrees of p i ( x ; ξ )as polynomials of x , and denoted by deg P . Definition 2.1.
For P ( x, ∂ ) ∈ W ( x, ξ ) of ord P > , we say that P ( x, ∂ ) is irreducible when it is satisfied that if we can write P = QR for some Q, R ∈ W ( x, ξ ) then ord Q · ord R = 0 . We recall some operations on W [ x, ξ ] and W ( x, ξ ). Details of theseoperations can be found in [11]. Definition 2.2 (The Fourier-Laplace transform) . The Fourier-Laplace trans-form on W [ x, ξ ] is the following algebra isomorphism L : W [ x, ξ ] −→ W [ x, ξ ] x ∂∂ xξ i ξ i ( i = 1 , . . . , n ) . Sometimes we identify P ∈ W ( x, ξ ) and f ( x ) P for f ( x ) ∈ C ( x, ξ ) be-cause differential equations P ( x, ∂ ) u = 0 and f ( x ) P ( x, ∂ ) u = 0 can be iden-tified. If for P, Q ∈ W ( x, ξ ), there exist f ( x ) ∈ C ( x, ξ ) such that P = f ( x ) Q ,then we write P ∼ Q. Definition 2.3 (The reduced representative) . The reduced representativeof a nonzero element P ∈ W [ x, ξ ] is an element of C ( x, ξ ) P ∩ W [ x, ξ ] of theminimal degree and denoted by R P . Definition 2.4.
For h ∈ C ( x, ξ ) , we define the automorphism of W ( x, ξ ) by Adei( h ) : W ( x ; ξ ) −→ W ( x ; ξ ) x x∂ ∂ − hξ i ξ i ( i = 1 , . . . , n ) . efinition 2.5 (The addition at the singular point) . For c ∈ C and f ( ξ ) ∈ C ( ξ ) we call the operator Adei( f ( ξ ) x − c ) the addition at the singular point c anddenote it by Ad(( x − c ) f ( ξ ) ) . We also define RAd(( x − c ) f ( ξ ) ) = R ◦ Ad(( x − c ) f ( ξ ) ) . Remark 2.6.
Let f ( x ) and g ( x ) be sufficiently many differentiable func-tions. Then the Leibniz rule tells us that ddx f ( x ) g ( x ) = ( f ( x ) ddx + ddx f ( x )) g ( x ) . Hence for some c, λ ∈ C we have ( x − c ) λ ddx ( x − c ) − λ g ( x ) = ( ddx − λx − c ) g ( x ) . Therefore
Adei( λx − c ) ddx corresponds to ( x − c ) λ ddx ( x − c ) − λ . Definition 2.7 (The e p ( x ) -twisting) . For p ( x ) ∈ C [ x ] , we define Ade( p ( x )) = Adei( p ′ ( x )) and call this the e p ( x ) -twisting. Here p ′ ( x ) = ddx p ( x ) . Remark 2.8.
As well as additions at singular points, the e p ( x ) -twistingcorresponds to the operation ddx e p ( x ) ddx e − p ( x ) = ( ddx − p ′ ( x )) . Definition 2.9 (The twisted Euler transform) . For α ∈ C and f ( ξ ) ∈ C ( ξ ) we define the operator on W [ x, ξ ] , E ( α, f ( ξ )) = L ◦
RAd(( x + α ) − f ( ξ ) ) ◦ L − ◦ R . For α = ( α , . . . , α m ) ∈ C m and F ( ξ ) = ( f ( ξ ) , . . . , f m ( ξ )) ∈ ( C ( ξ )) m , wedefine the operator E ( p ( x ); α, F ( ξ )) = Ade( p ( x )) ◦ m Y i =1 E ( α i , f i ( ξ )) ◦ Ade( − p ( x )) . We call E ( α, f ( ξ )) and E( p ( x ); α, F ( ξ )) twisted Euler transforms. In par-ticular we call E(0 , f ( ξ )) an Euler transform. emark 2.10. A twisted Euler transform E ( α, f ( ξ )) corresponds to thefollowing integral transform Z i ∞− i ∞ ( y + α ) − f ( ξ ) Z ∞ c g ( x ) e − xy dx e zy dy = 1Γ( f ( ξ )) Z zc g ( x )( z − x ) f ( ξ ) − e − α ( z − x ) dx. If we put α = 0, then this is a Riemann-Liouville integral. Hence E (0 , f ( ξ )) can be seen as the classical Euler transform. Remark 2.11.
A twisted Euler transform can be written as the compositionof e p ( x ) -twistings and an Euler transform. We define the parallel displace-ment for α ∈ C by P( h ) : W ( x ; ξ ) −→ W ( x ; ξ ) x x − α∂ ∂ξ i ξ i ( i = 1 , . . . , n ) . Then we can see that L − Ade( αx ) = P( − α ) L − and Ade( − αx ) L = L R( − α ) . Hence we have E ( α, f ( ξ )) = Ade( αx ) E (0 , f ( ξ ))Ade( − αx ) . Proposition 2.12.
We take an element P ( x, ∂ ) ∈ W ( x, ξ ) and assume that P ( x, ∂ ) is irreducible and ord P > . Then we have followings.1. We have E ( ∗ , P ∼ P where we can put any α ∈ C into ∗ .2. If E ( α, f ( ξ )) P is irreducible and ord E ( α, f ( ξ )) P > for an α ∈ C and an f ( ξ ) ∈ C ( ξ ) , we have E ( α, − f ( ξ )) E ( α, f ( ξ )) P = E ( α, P ∼ P . Before proving this proposition, we see the following lemma.
Lemma 2.13.
Suppose that P ∈ W [ x, ξ ] is irreducible. If E ( ∗ , P = g ( x ) ∈ C [ x, ξ ] , then there exists f ( x ) ∈ C [ x, ξ ] such that P ∼ f ( ∂ ) g ( x ) and the degree of f ( x ) as the polynomial of x is at most one. roof. By the assumption, we have E ( ∗ , P = L R L − R P = g ( x ) . This implies that g ( ∂ ) = L − g ( x ) = R L − R P, that is, there exists an f ( x ) ∈ C [ x, ξ ] such that L − R P = f ( x ) g ( ∂ ) . Equivalently we have P ∼ f ( − ∂ ) g ( x ) . By the irrducibility of P , f ( x ) must be an irreducible polynomial, i.e., itsdegree is at most one. proof of Proposition 2.12. Let us put Q = E ( ∗ , P = L R L − R P . Then we haveR L − R P = L − Q. Hence there exists an f ( x ) ∈ C [ x, ξ ] such that L − R P = f ( x ) L − Q, that is, we have P ∼ f ( − ∂ ) Q. By Lemma 2.13 and the assumption ord
P >
1, we can see that ord Q = 0.Since P is irreducible, we have f ( ∂ ) ∈ C . Hence we obtain E ( ∗ ; 0) P ∼ P. Let us recall thatRAd(( x + α ) f ( ξ ) ) ◦ RAd(( x + α ) − f ( ξ ) ) Q ∼ Q for Q ∈ W [ x, ξ ].By the assumption, E ( α, f ( ξ )) P is irreducible and ord E ( α, f ( ξ )) P > E ( α, − f ( ξ )) E ( α, f ( ξ )) = E ( α, − f ( ξ )) E ( ∗ , E ( α, f ( ξ )) P = L − RAd(( x + α ) f ( ξ ) )RAd(( x + α ) − f ( ξ ) ) L ◦ R P ∼ P. Formal solutions of differential equations
We study formal solutions of differential equations with Laurent series coef-ficients in this section. Although this section will contain many well-knownfacts, we will give proofs of these for the completeness of the paper. One ofthe purpose of this section is to understand when a system of fundamentalsolutions has no logarithmic singularities even though differences of charac-teristic exponents are integers. One of the answer of this question is givenby Oshima for Fuchsian differential equations in [11]. We apply his resultto formal solutions of differential equations with Laurent series coefficients.
Let K be the field of Laurent series of x − , also write C [[ x − ]][ x ], and Kh ∂ i the ring of differential operators with coefficients in K .Let us consider an element in Kh ∂ i , P ( x, ∂ ) = a n ( x ) ∂ n + a n − ( x ) ∂ n − + . . . + a ( x ) ∂ + a ( x )for a i ( x ) ∈ C [[ x − ]][ x ]. For a ( x ) = P ∞ n = −∞ a n x − n ∈ C [[ x − ]][ x ], v ( a ( x ))denote the valuation of a ( x ), i.e., v ( a ( x )) = min { s ∈ Z | a s = 0 } . Weconsider “normal” formal solutions near x = ∞ , i.e., solutions of the form x µ f ( x ) for some µ ∈ C and f ( x ) ∈ C [[ x − ]] ( f ( ∞ ) = 0).The group ring generated by { x µ | µ ∈ C } is denoted by C [ x µ ]. Alsothe polynomial ring of log x − with coefficients in C [[ x − ]] is denoted by C [[ x − ]][log x − ]. We consider the tensor product C [ x µ ] ⊗ C C [[ x − ]][log x − ]as C -algebras and we simply write these elements x µ ⊗ h ( x ) by x µ h ( x ) for µ ∈ C and h ( x ) ∈ C [[ x − ]][log x − ].As an analogue of systems of fundamental solutions around regular sin-gular points, we consider subspaces of C [ x µ ] ⊗ C C [[ x − ]][log x − ] which arespanned by formal power series with logarithmic terms. Definition 3.1.
Let us take µ , . . . µ r ∈ C , such that µ i − µ j / ∈ Z for i = j, increasing sequences of positive integers n i < · · · < n il i for i = 1 , . . . , r and m ij ∈ Z ≥ for i = 1 , . . . , r, j = 0 , . . . , l i . We put h ij = P jk =1 m ik and n = P ri =1 h il i .Then we consider the following n functions, f ( µ i + n ij ; k )( x ) = x µ i + n ij h ij + k X l =0 (cid:18) h ij + kl (cid:19) c ijh ij + k − l ( x )(log x − ) l , where (cid:18) kl (cid:19) are binomial coefficients, c ijk ( x ) ∈ C [[ x − ]] and c i ( ∞ ) = 0 . Wecall the C -vector subspace spanned by these functions the space of formalregular series. emma 3.2. We use the same notations as in Definition 3.1. Let V bea n -dimensional space of formal regular series. For i = 1 , . . . , r , we define C -linear maps Φ i : V −→ C [ x µ ] ⊗ C C [[ x − ]][log x − ] f ( x ) ∂ f ( x ) f ( µ i ;0)( x ) . Then each image Φ i ( V ) is n − -dimensional space of formal regular series.Proof. We can see that ∂f ( µ i + n ij ; k )( x ) = x µ i + n ij − h ij + k X l =0 (cid:18) h ij + kl (cid:19) H ijh ij + k − l ( x )(log x − ) l , where H ijh ij + k − l = { (( µ i + n k +1 ) + x∂ ) c ijh ij + k − l ( x ) − ( h ij + k − l ) c ijh ij + k − l − ( x ) } . Also we can see that ∂ f ( µ i ′ ; 0)( x ) = x − µ i ′ − F i ′ ( x ) , where F i ′ ( x ) = (( − µ i ′ + x∂ ) c i ′ ( x ) − ) . Clearly F i ′ ( ∞ ) = 0.Hence we have ∂ ( f ( µ i + n ij ; k )( x ) f ( µ i ′ ; k )( x ) ) = x µ i − µ i ′ + n ij − h ij + k X l =0 (cid:18) h ij + kl (cid:19) G ijh ij + k − l ( x )(log x − ) l . where G ijk ( x ) = H ijk ( x ) f ( µ i ′ ; 0)( x ) − + c ijk ( x ) F i ′ ( x ) . The image Φ i ′ ( V ) is spanned by (cid:26) ∂ ( f ( µ i + n ij ; k )( x ) f ( µ i ′ ; k )( x ) ) (cid:27) i,j,k . Since G i ( x ) =(( µ i − µ i ′ ) + x∂ ) c i ( x ) c i ′ ( x ) − , we have G i ( ∞ ) = 0 if i = i ′ . If i = i ′ , G i ( x ) = 0. However, we have ∂ ( f ( µ i ′ ; 1)( x ) f ( µ i ′ ; 0)( x ) ) = ∂ ( c i ′ ( x ) c i ′ ( x ) − + log x − ) . Hence we obtain that G i ( x ) = (1 + x∂c i ′ ( x ) c i ′ ( x ) − ) and G i ( ∞ ) = 0.Hence Φ i ′ ( V ) is the space of formal regular series.12 emma 3.3. Let P ( x, ∂ ) ∈ Kh ∂ i be P ( x, ∂ ) = a n ( x ) ∂ n + a n − ( x ) ∂ n − + . . . + a ( x ) ∂ + a ( x ) , for a i ( x ) ∈ C [[ x − ]][ x ] . It can be written by P ( x, ∂ ) = ∞ X s x ρ − s P s ( ∂ ) , where ρ = max { v ( a i ( x )) | i = 0 , . . . , n } and P s ( x ) ∈ C [ x ] . Then the follow-ings are equivalent1. P ( j ) i (0) = ∂ j P i ( x ) | x =0 = 0 for i + j < l. v ( a l − i − ( x )) < ρ − i ( i = 0 , . . . , l − .Proof. The condition 1 is equivalent to that P ( x, ∂ ) is written by P ( x, ∂ ) = x ρ ∂ l P ′ ( ∂ )+ x ρ − ∂ l − P ′ ( ∂ )+ · · · + x ρ − l +1 ∂P ′ l − ( ∂ )+ x ρ − l P l ( ∂ )+ · · · . Then the assertion is obvious.
Proposition 3.4.
Let P ( x, ∂ ) ∈ Kh ∂ i be P ( x, ∂ ) = a n ( x ) ∂ n + a n − ( x ) ∂ n − + . . . + a ( x ) ∂ + a ( x ) , for a i ( x ) ∈ C [[ x − ]][ x ] . Also we write P ( x, ∂ ) = ∞ X s =0 x ρ − s P s ( ∂ ) where P s ( x ) ∈ C [ x ] and deg P s ≤ n for s = 0 , , , . . . . For l ≤ n , we assume that there exist an l -dimensional space of formalregular series V whose elements are solutions of the differential equation P ( x, ∂ ) u = 0 .Then we have P ( j ) i (0) = 0 for i + j < l, equivalently v ( a l − i − ( x )) < ρ − i ( i = 0 , . . . , l − . Proof.
We prove this by induction on l . We assume that P ( x, ∂ ) has aformal solution x µ h ( x ) for ν ∈ C and h ( x ) ∈ C [[ x − ]] ( h ( ∞ ) = 0). Then thecoefficient of x µ of P ( x, ∂ ) x µ h ( x ), is P (0) h ( ∞ ) . Hence P (0) = 0 . We assume that there exists a k -dimensional space V of formal regularseries whose elements are solutions of P ( x, ∂ ) u = 0. We take an elementof V without log term, i.e., an element φ ( x ) = x µ h ( x ) for µ ∈ C and h ( x ) ∈ C [[ x − ]] ( h ( ∞ ) = 0). We consider a differential operetor ˜ P ( x, ∂ ) =13 ( x ) − P ( x, ∂ ) φ ( x ). Then there exits Q ( x, ∂ ) ∈ Kh ∂ i such that ˜ P ( x, ∂ ) = Q ( x, ∂ ) ∂ because 1 = φ ( x ) − φ ( x ) is a solution of ˜ P ( x, ∂ ) u = 0. Hence if weconsider a C -linear mapΦ : V −→ C [ x µ ] ⊗ C C [[ x − ]][log x − ] f ( x ) ∂ f ( x ) φ ( x ) , elements in the image Φ( V ) are formal solutions of Q ( x, ∂ ). By Lemma 3.2,Φ( V ) is a k − V ) are formal solutions of Q ( x, ∂ ) u = 0 by the construction. Hence bythe hypothesis of the induction we have Q ji (0) = 0 for i + j < k − . If we write Q ( x, ∂ ) = n − X i =0 q i ( x ) ∂ i for q k ( x ) ∈ C [[ x − ]][ x ] ( k = 0 , . . . , n − v ( q k − i − ( x )) < ρ − i ( i = 0 , . . . , k −
2) (3.1)by Lemma 3.3. Also we can write P ( x, ∂ ) = n X i =0 p i ( x ) ∂ i for p i ( x ) ∈ C [[ x − ]][ x ] ( i = 0 , . . . , n ). If we recall the Leibniz rule, ∂ i f ( x ) g ( x ) = i X j =0 (cid:18) ij (cid:19) f ( j ) ( x ) ∂ i − j g ( x ) , the equation φ ( x ) P ( x, ∂ ) φ ( x ) − = Q ( x, ∂ ) ∂ says that p n − i ( x ) = i X j =0 (cid:18) n − ji − j (cid:19) φ ( i − j ) ( x ) φ ( x ) q n − j − ( x ) ( i = 0 , . . . , n ) . Here we put q − ( x ) = 0. Since v ( φ ( i ) φ ( x ) ) ≤ − i , valuations of p i ( x ) are boundedas follows, v ( p i ( x )) ≤ max { v ( q j − ) − ( j − i ) | j = i, . . . , n } . If we notice that v ( p i +1 ( x )) ≤ max { v ( q j − ) − ( j − i ) + 1 | j = i + 1 , . . . , n } , v ( p i ( x )) ≤ max { v ( p i +1 ( x )) − , v ( q i − ( x )) } ( i = 1 , . . . , n ) . Then above inequalities and (3.1) implies that v ( p k − i − ( x )) < ρ − i ( i = 0 , . . . , k − . (3.2) Definition 3.5 (The characteristic equation) . Let us take an element of Kh ∂ i , P ( x, ∂ ) = ∞ X s =0 x ρ − s P s ( ∂ ) where P s ( x ) ∈ C [ x ] and deg P s ≤ n for s = 0 , , , . . . . We consider followingpolynomials of λ , p k ( λ, x ) = k X i =0 ( λ + i + 1) k − i k − i )! P ( k − i ) i ( x ) for k = 1 , . . . , n . Here we put ( λ ) l = λ ( λ + 1) · · · ( λ + ( l − for l = 1 , , . . . and ( λ ) = 1 . If P ( k )0 (0) = 0 , then p k ( λ, is a polynomial of λ of degree k .Hence if P ( m )0 (0) = 0 and P ( i ) j (0) = 0 for i + j < m , we call p m ( λ − m,
0) = 0 the characteristic equation of P ( x, ∂ ) . Lemma 3.6.
Let us take P ( x, ∂ ) as in Definition 3.5. Suppose that P ( j ) i (0) =0 for i + j < m and P ( m )0 (0) = 0 for an m ≤ n .Also we assume that there exist µ , . . . µ r ∈ C , such that µ i − µ j / ∈ Z for i = j, increasing sequences of positive integers n i < · · · < n il i for i = 1 , . . . , r and m ij ∈ Z > for i = 1 , . . . , r, j = 0 , . . . , l i such that m = P ri =1 P l i j =0 m ij . Also assume that solutions of the characteristic equation p m ( λ − m,
0) = 0 are µ i + n ij with multiplicities m ij for i = 1 , . . . , r , j = 0 , . . . , l i .Then there exist following m formal solutions of P ( x, ∂ ) u = 0 , f ( µ i + n ij ; k )( x ) = x µ i + n ij h ij + k X l =0 (cid:18) h ij + kl (cid:19) c ijh ij + k − l ( x )(log x ) l , where c ijk ( x ) ∈ C [[ x − ]] , c i ( ∞ ) = 0 and h ij = P jk =1 m ik . roof. Suppose that x µ P ∞ s =0 c x x − s ( c = 0) is a formal solution of P ( x, ∂ ) u = P ∞ s =0 x ρ − s P s ( ∂ ) u = 0. Then we have the following equations, i X k =0 c i − k p k ( µ − i,
0) = 0 , (3.3)for i = 0 , , . . . . The assumption implies p k ( λ,
0) for k < m are identicallyzero and p m ( λ,
0) is nonzero polynomial of λ of degree m . Hence this is adirect consequence of the Frobenius method. Proposition 3.7.
Let us take P ( x, ∂ ) ∈ Kh ∂ i as in Definition 3.5. Supposethat P ( j ) i (0) = 0 for i + j < m and P ( m )0 (0) = 0 . We take µ , . . . , µ r ∈ C such that µ i − µ j / ∈ Z ( i = j ) . Then the followings are equivalent,1. There exist m formal series, f ij ( x ) = x − µ i − j + x − µ i − m i h ij ( x ) where h ij ( x ) ∈ C [[ x − ]] for i = 1 , . . . , l and j = 0 , . . . , m i − and theseare solutions of P ( x, ∂ ) u = 0 .2. p k ( − µ i − j − k ; 0) = 0 for i = 1 , . . . , r,j = 0 , . . . , m i − ,k = m, m + 1 , . . . , m + m i − j − . Proof.
First we assume that 1 is true. We consider the equation P ( x, ∂ ) x − µ i − j ( c + ∞ X s = m i − j c s x − s ) = 0where i ∈ { , . . . , r } , j ∈ { , . . . , m i − } and c = 0. Since c s = 0 for1 < s < m i − j , the equation (3.3) tells us that c p k ( − µ i − j − k ; 0) = 0 , for m ≤ k < m i − j + m . If the condition 2 is false, p k ( − µ i − j − k ; 0) = 0for some k . This contradicts c = 0.Conversely we assume that 2 is true. Let us consider the equation P ( x, ∂ ) x − µ i − j ( c + P ∞ s = m i − j c s x − s ) = 0. Then the equation (3.3) implies c p k ( − µ i − j − k ; 0) = 0 , n ≤ k < m i − j + n and k X l =0 c m i − j + l p m +( k − l ) ( − µ i − m i − k − m, c p m i − j + m + k ( − µ i − m i − k − m,
0) = 0for k = 0 , , . . . . By the assumption 2, we can choose c = 0. Then if c is determined, other coefficients c s for s = m i − j, m i − j + 1 , . . . aredetermined inductively. We can put c = 1. Then we can choose x − µ i − j (1 + P ∞ s = m i − j c s x − s ) as a formal solution of P ( x, ∂ ) f ( x ) = 0 for all i ∈ { , . . . , r } , j ∈ { , . . . , m i − } . Definition 3.8 (Semi-simple characteristic exponents) . Let us take a dif-ferential operator P ( x, ∂ ) = ∞ X s =0 x ρ − s P s ( ∂ ) where P s ( x ) ∈ C [ x ] and deg P s ≤ n for s = 0 , , , . . . . Suppose that P ( j ) i (0) =0 for i + j < m and P ( m )0 (0) = 0 . Also suppose that there exit µ , . . . , µ r ∈ C such that µ i − µ j / ∈ Z ( i = j ) and these satisfy p k ( − µ i − j − k,
0) = 0 for i = 1 , . . . , r,j = 0 , . . . , m i − ,k = m, m + 1 , . . . , m + m i − j − . Here m = P ri =1 m i . Then we say that P ( x, ∂ ) has semi-simple characteristicexponents { µ , µ + 1 , . . . , µ + m − , . . . , µ r , µ r + 1 , . . . , µ r + m r − } , at x = ∞ . By using the notation [ µ ] m = { µ, µ + 1 , . . . µ m − } , we write { [ µ ] m , . . . , [ µ r ] m r } = { µ , µ + 1 , . . . , µ + m − , . . . , µ r , µ r + 1 , . . . , µ r + m r − } shortly. Let us recall that e − p ( x ) ∂e p ( x ) = ∂ + p ′ ( x ) . P ( x, ∂ ) ∈ Kh ∂ i , the differential equation P ( x, ∂ ) u = 0 has a formalsolution e p ( x ) x − ν X i =0 c s x − s if and only if the differential equation P ( x, ∂ + p ′ ( x )) has a formal solution x − ν X i =0 c s x − s . Definition 3.9 ( e p ( x ) -twisted semi-simple characteristic exponents) . For P ( x, ∂ ) ∈ Kh ∂ i and p ( x ) ∈ C [ x ] , we say that the differential equation P ( x, ∂ ) u = 0 has e p ( x ) -twisted semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ where µ i ∈ C , m i ∈ N and m = P li =1 m i , if the differentialequation P ( x, ∂ + p ′ ( x )) has the same semi-simple exponents at x = ∞ . Proposition 3.10.
Suppose that P ( x, ∂ ) ∈ W [ x ] has e p ( x ) -twisted semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ where µ i ∈ C , m i ∈ N and m = P li =1 m i .1. For α, ν ∈ C , the differential equation Ad(( x − α ) ν ) P ( x, ∂ ) u ( x ) = 0 has e p ( x ) -twisted semi-simple exponents { [ µ − ν ] m , . . . , [ µ l − ν ] m l } at x = ∞ .2. For q ( x ) ∈ C , the differential equation Ade( q ( x )) P ( x, ∂ ) has e p ( x )+ q ( x ) -twisted semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ .Proof. If we recall that ( x − α ) ν can be written as ( x − α ) ν = x − ν P ∞ i =0 c i x − i ,the first assertion follows from the same argument as Proposition 3.14. Thesecond assertion easily follows fromAde( q ( x )) P ( x, ∂ ) = exp( q ( x )) P ( x, ∂ ) exp( − q ( x )) . .2 A review of regular singularity Let us take a differential operator P ( x, ∂ ) ∈ W [ x ]. If P ( x, ∂ ) has a regularsingular point at x = c ∈ C , then we can write P ( x, ∂ ) = n X i =0 ( x − c ) n − i a i ( x ) ∂ n − i where a ( c ) = 0. We consider a polynomial of νf c ( x, ν ) = n X j =0 a j ( x ) a ( x ) ν ( ν − · · · ( ν − ( n − j ) + 1) . Since x = c is a regular singular point of P ( x, ∂ ), f c ( x, ν ) is holomorphic at x = c . Hence we have the Taylor expansion f c ( x, ν ) = ∞ X k =0 f ck ( ν )( x − c ) k , (3.4)where f ck ( ν ) are polynomials of ρ . Then a power series g ( ν, x ) = ( x − c ) ν P ∞ k =0 d k ( x − c ) k satisfies P ( x, ∂ ) g ( ν, x ) = 0 if and only if equations l X k =0 c l − k f ck ( ν + ( l − k )) = 0 (3.5)are satisfied for l = 0 , , . . . . We call the equation f c ( ρ ) = 0 the characteristicequation at regular singular point x = c .Then we have a similar result to Proposition 3.7. Proposition 3.11 (Oshima [11]) . Let us take P ( x, ∂ ) ∈ W [ x ] which has aregular singular point x = c . We write P ( x, ∂ ) = n X i =0 ( x − c ) n − i a i ( x ) ∂ n − i where a i ( x ) ∈ C [ x ] and a ( c ) = 0 . Then we can define polynomials f ck ( ν ) for k = 0 , , , . . . as ( ) . Then the followings are equivalent.1. There exist µ , . . . , µ l ∈ C such that µ i − µ j / ∈ Z if i = j . The following n functions are solutions of P ( x, ∂ ) u = 0 , g ij ( x ) = ( x − c ) µ i + j + ( x − c ) µ i + m i h ij ( x − c ) for i = 1 , . . . , l and j = 0 , . . . , m i − . Here h ij ( x ) ∈ C [[ x ]] and n = P li =1 m i . . There exist µ , . . . , µ l ∈ C such that µ i − µ j / ∈ Z if i = j . For these µ i , we have f k ( µ i − j ) = 0 for i = 1 , . . . , l , j = 0 , . . . , m i − and k = 0 , . . . , m i − j − . Proposition 3.12 (Oshima [11]) . Let us take P ( x, ∂ ) ∈ W [ x ] which has aregular singular point x = c . We write P ( x, ∂ ) = n X i =0 ( x − c ) n − i a i ( x ) ∂ n − i where a i ( x ) ∈ C [ x ] and a ( c ) = 0 . Then the followings are equivalent.1. There exist m functions, g i ( x ) = ( x − c ) i + ( x − c ) m h i ( x − c ) for i = 0 , . . . , m − and these are solutions of P ( x, ∂ ) u = 0 . Here h i ( x ) ∈ C [[ x ]] .2. There exist Q ( x, ∂ ) ∈ W [ x ] such that P ( x, ∂ ) = ( x − c ) m Q ( x, ∂ ) . Proof.
Although the proof of this proposition can be found in [11], we provethis for the completeness. Suppose that 2 is true. We notice that Q ( x, ∂ )( x − c ) i are holomorphic at x = c for any i ∈ Z ≥ . Hence if we write Q ( x, ∂ )( x − c ) i = h i ( x ) ∈ C [[ x ]], then we have P ( x, ∂ )( x − c ) i = ( x − c ) m Q ( x, ∂ )( x − c ) i = ( x − c ) m h i ( x ) . Conversely, if 1 is true, there exist h i ( x ) ∈ C [[ x ]] such that P ( x, ∂ )( x − c ) i = n X j =0 ( x − c ) n − j a j ( x ) ∂ n − j ( x − c ) i = ( x − c ) m h i ( x ) , for i = 0 , . . . , m −
1. If i = 0, we have a ( x ) = ( x − c )(( x − c ) m − h ( x ) − n X j =1 ( x − c ) n − i − a j ( x )) . Hence P ( x, ∂ ) = ( x − c ) Q ( x, ∂ ) for a Q ( x, ∂ ) ∈ W [ x ]. If i = 1, we have( x − c )( a ( x ) + a ( x )) = ( x − c ) (( x − c ) m − − n X j =2 ( x − c ) n − j − a j ( x )) . Hence P ( x, ∂ ) = ( x − c ) Q ( x, ∂ ) for a Q ( x, ∂ ) ∈ W [ x ] . We can iteratethese for i = 0 , , . . . , m −
1. 20e can define semi-simple characteristic exponents at regular singularpoints as we define for formal solutions.
Definition 3.13.
Let us take P ( x, ∂ ) ∈ W [ x ] which has a regular singularpoint x = c . We write P ( x, ∂ ) = n X i =0 ( x − c ) n − i a i ( x ) ∂ n − i where a i ( x ) ∈ C [ x ] and a ( c ) = 0 . Then we can define polynomials f ck ( ν ) for k = 0 , , , . . . as ( ) . If there exist µ , . . . , µ l ∈ C such that µ i − µ j / ∈ Z ( i = j ) and we have f k ( µ i − j ) = 0 for i = 1 , . . . , l , j = 0 , . . . , m i − and k = 0 , . . . , m i − j − , then we saythat P ( x, ∂ ) has semi-simple characteristic exponents { [ µ ] m , . . . , [ µ l ] m l } at x = c . Proposition 3.14.
Let P ( x, ∂ ) ∈ W [ x ] has a regular singular point x = c and semi-simple exponents { [ ρ ] m , . . . , [ ρ l ] m l } .
1. For ν ∈ C , the differential equation Ad(( x − c ) ν ) P ( x, ∂ ) u = 0 hassemi-simple exponents { [ ρ + ν ] m , . . . , [ ρ l + ν ] m l } at x = c .2. If α = c , then the addition at x = α does not change the set of expo-nents at x = c of P ( x, ∂ ) , that is, Ad(( x − α ) ν ) P ( x, ∂ ) has semi-simpleexponents { [ ρ ] m , . . . , [ ρ l ] m l } at x = c as well.3. For p ( x ) ∈ C , the set of exponents at x = c of P ( x, ∂ ) are not changedby Ade( p ( x )) , that is, Ade( p ( x )) P ( x, ∂ ) has semi-simple exponents { [ ρ ] m , . . . , [ ρ l ] m l } at x = c as well. roof. If a function u ( x ) is a solution of P ( x, ∂ ) u ( x ) = 0, then for α, ν ∈ C ,the function ( x − α ) ν u ( x ) satisfies thatAd(( x − α ) ν ) P ( x, ∂ )( x − α ) ν u ( x )= ( x − α ) ν P ( x, ∂ )( x − α ) − ν ( x − α ) ν u ( x )= ( x − α ) ν P ( x, ∂ ) u ( x ) = 0 . Hence if α = c and u ( x ) = ( x − c ) ρ P ∞ i =0 c i x i is a solution of P ( x, ∂ ) u ( x ) = 0around x = c , then ( x − c ) ν u ( x ) = ( x − c ) ρ + ν P ∞ i =0 d i x i is a solution ofAd(( x − c ) ν ) P ( x, ∂ ) v ( x ) = 0.On the other hand, if α = c , the function ( x − α ) ν is holomorphic at x = c . Hence we can write the Taylor expansion ( x − α ) ν = P ∞ i =0 e i x i . Thisimplies that ( x − α ) ν u ( x ) = P ∞ i =0 e i x i ( x − c ) ρ P ∞ j =0 c j x i = ( x − c ) ρ P ∞ i =0 f i x i .Therefore exponents does not change.For the final assertion, we recall thatAde( p ( x )) P ( x, ∂ ) = exp( p ( x )) P ( x, ∂ ) exp ( − p ( x )) , and exp(( p ( x )) is holomorphic on C . By the same argument as above, thefinal assertion follows. at in-finity The twisted Euler transform turns P ( x, ∂ ) ∈ W [ x, ξ ] into the other Q ( x, ∂ ) ∈ W [ x, ξ ]. The question is how this transformation changes local datum ofthese differential operators. We focus on the case of the rank of irregularsingularity at most 2 and give explicit descriptions about the changes ofcharacteristic exponents by twisted Euler transforms. Let us recall the notions, the rank of irregular singularity and the Newtonpolygon of a differential operator.
Definition 4.1 (The rank of irregular singularity at infinity) . Let us con-sider a linear differential equation, [ x n ∂ n + a ( x ) x n − ∂ n − + a ( x ) x n − ∂ n − + · · · + a n ( x )] f ( x ) = 0 . (4.1) Here coefficients a i ( x ) are Laurent series, a i ( x ) = x m i ∞ X k =0 a ik x − k ( a i = 0) , here m i ∈ Z for i = 1 , . . . , n .The rank of irregular singularity at infinity of ( ) is the number definedby q = max { m i i | i = 1 , . . . , n } . Let us take P ( x, ∂ ) = P a i ( x ) ∂ i ∈ Kh ∂ i . Every a i ( x ) ∂ i associate thepoint ( i, i − v ( a i )) of N × Z . Then we define the Newton polygon N ( P ) of P to be the convex hull of the set [ i { ( x, y ) ∈ R | x ≤ i, y ≥ i − v ( a i ) } . Let { s i = ( u i , v i ) } ≤ i ≤ p be the set of vertices of this polygon such that0 = u < u < · · · < u p = n ( n = ord P ). Slopes of the edge connecting s i and s i − are λ i = v i − v i − u i − u i − for i = 0 , . . . , p −
1. Clearly we have λ < λ < · · · < λ p . We define lengths L i of segments [ s i − , s i ] by L i = u i − u i − . We note that λ p corresponds tothe irregular rank at infinity of P . We refer [10],[12] and [14] for furtherthings about Newton polygons. Remark 4.2.
Let us consider the Newton polygon of P ( x, ∂ ) whose ver-tices s , s , . . . , s r , slopes λ < λ < · · · < λ r and lengths of segments are L , . . . , L r . Then for ≤ i ≤ r , there exist q ∈ Z > and L i linearly indepen-dent formal solutions of P ( x, ∂ ) u = 0 , f ijk ( x ) = e p ij ( x ) x µ ijk h ijk ( x q ) for q ∈ Z > h ijk ( x ) ∈ C [[ x − ]][log x − ] . Here p ij ( x ) = r ij X l =0 a ij ( l ) x ri − lq and r i q = λ i . As is well-known, the Fourier-Laplace transform exchanges regular singularpoints on C and irregular singular point of rank 1 at infinity. Let us see theway how the Fourier-Laplace transform changes ranks of irregular singulari-ties and characteristic exponents of a differential operators of irregular rankat most 2. Proposition 4.3.
Let us take P ( x, ∂ ) ∈ W [ x ] of deg P = N and µ , . . . , µ l ∈ C such that µ i / ∈ Z and µ i − µ j / ∈ Z ( i = j ) . Then the followings are equiv-alent. . The differential equation P ( x, ∂ ) u = 0 has semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ . Here m = P li =1 m i .
2. For the Fourier-Laplace transform P ( − ∂, x ) of P ( x, ∂ ) has regular sin-gular point at x = 0 and P ( − ∂, x ) u = 0 has the semi-simple charac-teristic exponents, { [0] N − m [ µ − m , . . . , [ µ l − m l } at x = 0 .Proof. Suppose that 1 is true. From the assumption µ i / ∈ Z , we can see N ≥ m . If N < m , we can write P ( x, ∂ ) = N X s =0 x N − s P s ( ∂ ) ∂ m − s = Q ( x, ∂ ) ∂ N − m for Q ( x, ∂ ) ∈ W [ x ]. Then polynomials P N − m − i =0 a i x for a i ∈ C satisfy P ( x, ∂ ) u = 0. It contradicts our assumption.If we write P ( x, ∂ ) = P Ns =0 x N − s P s ( ∂ ) ∂ max { m − s, } , the Laplace trans-form P ( − ∂, x ) = N X s =0 ( − ∂ ) N − s P s ( x ) x max { m − s, } = N X s =0 Q s ( x ) x max { m − s, } ( − ∂ ) N − s for P s ( x ) , Q s ( x ) ∈ C [ x ] and P ( x ) = Q ( x ). By the assumption P (0) = Q (0) = 0, it follows that x = 0 is a regular singular point of P ( ∂, x ) u = 0.Let us take a power series x µ P ∞ s =0 d s x s . Then we can see that L P ( x, ∂ ) x µ ∞ X s =0 d s x s = N X j =0 ( − ∂ ) N − j P j ( x ) x µ ∞ X s =0 d s x s = ∞ X s = m x µ − N + s s X k = m d s − k k X l =0 ( − µ − s + l )( − µ − s + l +1) · · · ( − µ − s + N − P ( k − l ) l (0)( k − l )!= ∞ X s = m x µ − N + s s X k = m d s − k k X l =0 ( − µ − s + l ) ( k − l ) ( − µ − s + k ) ( N − k ) P ( k − l ) l (0)( k − l )!= ∞ X s = m x µ − N + s s X k = m ( − µ − s + k ) ( N − k ) d s − k p k ( − µ − , . L P ( x, ∂ ) x µ P ∞ s =0 d s x s = 0, then it must be satisfied that s X k = m d s − k ( − µ − s + k ) ( N − k ) p k ( − µ − − s, s = m, m + 1 , . . . . By the assumption we have p k ( − µ i − j − k,
0) = 0for i = 1 , . . . , l,j = 0 , . . . , m i − ,k = m, m + 1 , . . . , m + m i − j − . Also we have ( − i − s + k ) ( N − k ) = 0for i = 0 , . . . , ( N − m ) − k = m, m + 1 , . . . , N − j −
1. Hence thedifferential equation L P ( x, ∂ ) u = 0 has a regular singular point at x = 0and semi-simple characteristic exponents { [0] N − m , [ µ − m , . . . , [ µ l − m l } . The converse direction can be shown by the same way.The same thing can be shown for the Fourier-Laplace inverse transform.
Proposition 4.4.
Let us take P ( x, ∂ ) ∈ W [ x ] of deg P = N and µ , . . . , µ l ∈ C such that µ i / ∈ Z and µ i − µ j / ∈ Z ( i = j ) . Then the followings are equiv-alent.1. The differential equation P ( x, ∂ ) u = 0 has semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ . Here m = P li =1 m i .
2. For the Fourier-Laplace inverse transform P ( ∂, − x ) of P ( x, ∂ ) has aregular singular point at x = 0 and P ( ∂, − x ) u = 0 has the semi-simplecharacteristic exponents, { [0] N − m [ µ − m , . . . , [ µ l − m l } at x = 0 . roof. The condition 1 is equivalent to that P ( − x, − ∂ ) has semi-simpleexponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ . This is equivalent to that P ( ∂, − x ) has semi-simple exponents { [0] N − m [ µ − m , . . . , [ µ l − m l } at x = 0 by Proposition 4.3. Corollary 4.5.
Let us take P ( x, ∂ ) ∈ W [ x ] of deg P = N and µ , . . . , µ l ∈ C such that µ i / ∈ Z and µ i − µ j / ∈ Z ( i = j ) . Then the followings are equivalent.1. The differential equation P ( x, ∂ ) u = 0 has e αx -twisted semi-simpleexponents { [ µ ] m , . . . , [ µ l ] m l } .
2. For the Laplace transform P ( − ∂, x ) of P ( x, ∂ ) has a regular singularpoint at x = α and semi-simple exponents { [0] N − m [ µ − m , . . . , [ µ l − m l } at x = α . Here m = P li =1 m i . Proof.
The condition 1 is equivalent to that P ( x, ∂ + α ) u = 0 has semi-simpleexponents { [ µ ] m , . . . , [ µ l ] m l } at x = ∞ . Hence it is equivalent to that the Laplace transform P ( − ∂, x + α ) u = 0 has a regular singular point at x = 0, i.e., L P ( x, ∂ ) = P ( − ∂, x ) hasa regular singular point at x = α and semi-simple exponents { [0] N − n [ µ − m , . . . , [ µ l − m l } at x = α .For the inverse transform we can show the following as well. Corollary 4.6.
Let us take P ( x, ∂ ) ∈ W [ x ] of deg P = N and µ , . . . , µ l ∈ C such that µ i / ∈ Z and µ i − µ j / ∈ Z ( i = j ) . Then the followings are equivalent.1. The differential equation P ( x, ∂ ) u = 0 has e αx -twisted semi-simpleexponents { [ µ ] m , . . . , [ µ l ] m l } .
2. For the Fourier-Laplace inverse transform P ( ∂, − x ) of P ( x, ∂ ) has aregular singular point at x = − α and semi-simple exponents { [0] N − m [ µ − m , . . . , [ µ l − m l } at x = − α . Here m = P li =1 m i . roof. We can show this by the same argument as Corollary 4.5.
Corollary 4.7.
Let us take P ( x, ∂ ) ∈ W [ x ] of deg P = N and µ , . . . , µ l ∈ C such that µ i / ∈ Z and µ i − µ j / ∈ Z ( i = j ) . Then the followings are equivalent.1. The differential equation P ( x, ∂ ) u = 0 of has e α x + βx -twisted semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } .
2. The Laplace transform P ( − ∂, x ) u = 0 has e − α x + βα x -twisted semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } . Proof.
The condition 1 is equivalent to that P ( x, ∂ + αx + β ) u = 0 hassemi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } . at x = ∞ . On the other hand, the condition 2 is equivalent to that P ( − ∂ + α x − βα , x ) u = 0 has the same semi-simple exponents at infinity. If we put x = αy , it is equivalent to say that P ( y − α ∂ y − βα , αy ) v = 0 has the samesemi-simple exponents at infinity. Here ∂ y = ddy . It is easy to see that L ◦
Ade(( 12 α y − βα y )) ◦ L − P ( y − α ∂ y − βα , αy ) = P ( y, ∂ y + αy + β ) . If we notice that for a solution u of Q ( x, ∂ ) u = 0, v = e p ( x ) u is a solutionof Ade( p ( x )) Q ( x, ∂ ) v = 0. Since e p ( x ) is holomorphic at x = 0, the multipli-cation of e p ( x ) does not change exponents at x = 0. Then the equivalence 1and 2 follows from Proposition 4.3. Corollary 4.8.
Let us take P ( x, ∂ ) ∈ W [ x ] of deg P = N and µ , . . . , µ l ∈ C such that µ i / ∈ Z and µ i − µ j / ∈ Z ( i = j ) . Then the followings are equivalent.1. The differential equation P ( x, ∂ ) u = 0 has e α x + βx -twisted semi-simpleexponents { [ µ ] m , . . . , [ µ l ] m l } .
2. The Fourier-Laplace inverse transform P ( ∂, − x ) u = 0 has e − α x − βα x -twisted semi-simple exponents { [ µ ] m , . . . , [ µ l ] m l } . Proof.
The condition 2 is equivalent to that P ( ∂ − α x − βα , − x ) has the abovesemi-simple exponents at x = ∞ . If we put y = − α x , this is equivalent to P ( − α ∂ y + y − βα , αy ) has the same exponents at y = ∞ . Also we have L − ◦ Ade( 12 α y + βα ) ◦ L P ( − α ∂ y + y − βα , αy ) = P ( y, ∂ y + αy + β ) . As in Corollary 4.7, we can show this corollary.27 roposition 4.9.
Let us take P ( x, ∂ ) ∈ W [ x ] . We assume that P ( x, ∂ ) canbe written by R P ( x, ∂ ) = x N r Y i =0 ( ∂ − α i ) m i + N − X j =1 x N − j r Y i ( ∂ − α i ) max { m i − j, } P j ( ∂ ) for P i ( x ) ∈ C [ x ] and P ( x, ∂ ) has e α i x -twisted semi-simple exponents { [ µ i ] m i , . . . , [ µ il i ] m ili } where m i = P l i j =1 m ij . Moreover we assume that µ ij / ∈ Z for i = 1 , . . . , r and j = 1 , . . . , l i and µ ij − µ ik / ∈ Z ( k = j ) .Then we have the followings.1. We have E ( ∗ ; 0) P ( x, ∂ ) ∼ P ( x, ∂ ) for i = 1 , . . . , r .2. For fixed i ∈ { , . . . , r } and µ ∈ C such that µ / ∈ Z and µ ij − µ / ∈ Z \{ } for all j = 1 , . . . l i , we have E ( α i ; − µ ) E ( α i ; µ ) P ( x, ∂ ) ∼ P ( x, ∂ ) for i = 1 , . . . , r .Proof. First we show 1. We have E ( α i ; 0) P ( x, ∂ ) = L R L − R P ( x, ∂ )= L R( r Y i =1 ( − x − α i ) m i ∂ N + N X j =1 r Y i =1 ( − x − α i ) max { m i − j, } a j ( x ) ∂ N − j ) . By the assumption and Corollary 4.6, every characteristic exponent at theregular singular point x = − α i of L − R P ( x, ∂ ) is not integers. Hence byProposition 3.12, R L − R P ( x, ∂ ) = L − R P ( x, ∂ ). Hence E ( ∗ ; 0) P ( x, ∂ ) = L R L − R P ( x, ∂ ) ∼ P ( x, ∂ ) . Let us show 2. Fix an i ∈ { , . . . r } . If µ ij − µ = 1 for all j = 1 , . . . , l i , E ( α i , µ ) P ( x, ∂ ) has e α i x -twisted semi-simple exponents { [ − µ + 1] N − m i [ µ i − µ ] m i , . . . , [ µ il i − µ ] m ili } and e α i ′ x -twisted semi-simple exponents { [ µ i ′ ] m i ′ , . . . , [ µ i ′ l i ′ ] m i ′ li ′ } i ′ by corollaries from 4.5 to 4.8. On the other hand, if there exist j ∈ { , . . . , l i } such that µ ij − u , then E ( α i , µ ) P ( x, ∂ ) has e α i x -twistedsemi-simple exponents { [ µ i − µ ] m i , . . . , [ µ ij − − µ ] m ij − , [ − µ +1] N − m i , [ µ ij +1 − µ ] m ij +1 , . . . , [ µ il i − µ ] m ili } and e α i ′ x -twisted semi-simple exponents { [ µ i ′ ] m i ′ , . . . , [ µ i ′ l i ′ ] m i ′ li ′ } for the other i ′ by corollaries from 4.5 to 4.8 In both cases, characteristicexponents are not integers and moreover difference of them are not integers.Hence we can see E ( ∗ , E ( α i , µ ) P ∼ E ( α i , µ ) P by the same argument as 1. Hence we have 2 as well as the proof of Propo-sition 2.12. Definition 4.10 (Normal at infinity) . Let us take P ( x, ∂ ) ∈ W [ x, ξ ] whichhas an irregular singular point at x = ∞ of rank 2. The order of P ( x, ∂ ) is n . We say that P ( x, ∂ ) is normal at the irregular singular point x = ∞ ,if the followings are satisfied. There exist α i , β ij ∈ C for i = 1 , . . . , r and j = 1 , . . . . , l i .1. For every i = 1 , . . . , r , the Newton polygon of P ( x, ∂ + α i x ) has onlythree vertices s i = ( u i = 0 , v i ) , s i = ( u i , v i ) and s i = ( u i = n, v i ) .Corresponding slopes are and . Length of the segment [ s i , s i ] is n i .Here n = P ri =1 n i .2. For every ( i, j ) , i = 1 , . . . , r and j = 1 , . . . , l i , the Newton polygon of P ( x, ∂ + α i x + β ij ) has only four vertices s ij = ( u ij = 0 , v ij ) , s ij =( u ij , v ij ) , s ij = ( u ij , v ij ) , s ij = ( u ij = n, v ij ) . Corresponding slopes are , and . Lengths of segments [ s ij , s ij ] , [ s ij , s ij ] and [ s ij , s ij ] are n ij , n i − n ij and n − n i respectively. Here n i = P l i j =1 n ij . Remark 4.11.
By Proposition 3.4, Proposition 3.6 and Remark 4.2, P ( x, ∂ ) ∈ W [ x, ξ ] is normal at infinity if and only if there are n ij -dimensional space of e αi x + β ij x -twisted formal solutions. Since n = P ri =1 P l i j =1 n ij coincides withthe order of P ( x, ∂ ) , these are all of formal solutions of P ( x, ∂ ) u = 0 at x = ∞ . P ( x, ∂ ) ∈ W [ x, ξ ] which has regular singular points at c , . . . , c p ∈ C , the irregular singular point of rank 2 at x = ∞ and noother singular points. Then we can write P ( x, ∂ ) = n X i =0 p Y j =1 ( x − c j ) n − i a i ( x ) ∂ n − i , (4.2)where a i ( x ) ∈ C [ x, ξ ] for i = 1 , . . . , n and a ( x ) = 1. Since the rank ofirregularity is 2, degrees of a i ( x ) have upper bound,deg a i ( x ) ≤ ( p + 1) i. We call (4.2) the standard form of P ( x, ∂ ). Definition 4.12 (Table of local datum) . For sufficiently large numbers K and K ′ , we take ξ = { µ ij | ≤ i, j ≤ K } ∪ { ν ijk | ≤ i, j, k ≤ K ′ } as theindeterminants of C ( ξ ) . Let us take P ( x, ∂ ) ∈ W [ x, ξ ] which has regularsingular points at c , . . . , c p ∈ C , the irregular singular point of rank 2 at x = ∞ and no other singular points. Moreover P ( x, ∂ ) is normal at infinity.Let us assume that the differential equation P ( x, ∂ ) u = 0 has the follow-ing local solutions. • Around each regular singular point x = c i , it has semi-simple exponents { [0] m i [ µ i ] m i , . . . , [ µ is i ] m isi } where m ij ∈ Z > for i = 1 , . . . , s i and m i ∈ Z ≥ which satisfy P s i j =0 m ij = n = ord P for i = 1 , . . . , p . For the simplicity, we put µ i = ( µ i , . . . , µ is i ) , m i = ( m i , . . . , m is i ) , and write [ µ i ; m i ] = { [ µ i ] m i , . . . , [ µ is i ] m isi } , shortly. • Around irregular singular point x = ∞ , it has e αi x + β j x -twisted semi-simple exponents { [ ν ij ] n ij , . . . , [ ν ijt ij ] n ijtij } for i = 1 , . . . , r , j = 1 , . . . , l i . Here n ijk ∈ Z > and P ri =1 P l i j =1 P t ij k =1 n ijk = n = ord P ( x, ∂ ) . We put n ij = P t ij k =1 n ijk and n i = P l i j =1 n ij . For thesimplicity, we put ν ij = ( ν ij , . . . , ν ijt ij ) , n ij = ( n ij , . . . , n ijt ij ) , and write [ ν ij ; n ij ] = { [ ν ij ] n ij , . . . , [ ν ijt ij ] n ijtij } . hen we write the following table of local exponents of P ( x, ∂ ) , α α · · · α r c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] ... ... ... ... ... ... ... ... c p [ µ p ; m p ] β l [ ν l ; n l ] β l [ ν l ; n l ] β rl r [ ν rl r ; n rl r ] .We call this table the table of local datum of P ( x, ∂ ) .In particular, if P ( x, ∂ ) = ∂ + αx + β for some α, β ∈ C , we say P ( x, ∂ ) has the trivial table of local datum. Remark 4.13.
In Definition 4.12, the indeterminants ξ = { µ ij | ≤ i, j ≤ K } ∪ { ν ijk | ≤ i, j, k ≤ K ′ } have only one linear relation which comes fromFuchs relation (see [1] and [2] for example). Theorem 4.14.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12. Weassume that P ( x, ∂ ) has the following nontrivial table of local datum, α α · · · α r c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] ... ... ... ... ... ... ... ... c p [ µ p ; m p ] β l [ ν l ; n l ] β l [ ν l ; n l ] β rl r [ ν rl r ; n rl r ] .Changing the order of α , . . . , α r and applying Ade ( α x ) , We can assumethat α = 0 . Then L ◦ R P ( x, ∂ ) has ord L ◦ R P ( x, ∂ ) = p X i =1 s i X j =1 m ij + ( n − n ) , and the table of local datum, − α · · · − α r β [˜ ν ; n ] − c [˜ µ ; m ] β α [ ν ; n ] · · · β r α r [ ν r ; n r ] β [˜ ν ; n ] − c [˜ µ ; m ] β α [ ν ; n ] · · · β r α r [ ν r ; n r ] ... ... ... ... ... ... ... ... β l [˜ ν l ; n l ] − c p [˜ µ p ; m p ] β l α [ ν l ; n l ] β rlr α r [ ν rl r ; n rl r ] .Here ˜ µ i = ( µ i + 1 , µ i + 1 , . . . , µ is i + 1) , ˜ ν j = ( ν j − , ν j − , . . . , ν jt j − , for i = 1 , . . . , p and j = 1 , . . . , l . roof. By Proposition 3.11, the standard form of P ( x, ∂ ) can be dividedby φ ( x ) = Q pi =1 ( x − c i ) m i , i.e., there exist Q ( x, ∂ ) ∈ W [ x, ξ ] such that P ( x, ∂ ) = φ ( x ) Q ( x, ∂ ) and moreover R P ( x, ∂ ) = Q ( x, ∂ ). Since we assumethat α = 0, we can see deg P ( x, ∂ ) = ( p + 1)( n − n ). Hence deg Q ( x, ∂ ) =deg P ( x, ∂ ) − P pi =1 m i = ( p + 1)( n − n ) − P pi =1 m i = P pi =1 ( n − m i ) + ( n − n ) = P pi =1 P s i j =1 m ij + ( n − n ).Then this theorem is obtained by Corollary 4.5 and Corollary 4.7. Remark 4.15.
We can show the same thing as the above theorem for theFourier-Laplace inverse transform by Corollary 4.6 and Corollary 4.8.
Definition 4.16 (The rigidity index) . Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as inDefinition 4.12. Then we define the number idx P = − (( p + 1) n − r X i =1 n i − r X i =1 l i X j =1 ( n ij ) − p X i =1 s i X j =0 ( m ij ) − q X i =1 l i X j =1 t ij X k =1 ( n ijk ) ) , and call this the rigidity index of P . Remark 4.17.
The standard form of P ( x, ∂ ) is P ( x, ∂ ) = n X i =0 p Y j =1 ( x − c j ) n − i a i ( x ) ∂ n − i , where deg a i ( x ) ≤ ( p + 1) i. Hence P ( x, ∂ ) has n X i =1 (( p + 1) i + 1) coefficients in C ( ξ ) . The informations about the sets of exponents at regularsingular points x = c , . . . , c p require p X i =1 s i X j =0 m ij ( m ij + 1)2 linear equations by Lemma 3.11. On the other hand, P ( x, ∂ ) has n i -dimensionalformal solutions with e αi x + β ij x -twisted semi-simple exponents for i = 1 , . . . , r and j = 1 , . . . , l i . Hence the standard form of P ( x, ∂ + α i x ) is P ( x, ∂ + α i x ) = n X k =0 p Y l =1 ( x − c l ) n − k b ik ( x ) ∂ n − k , here ( deg b ik ( x ) ≤ ( p + 1) k for ≤ k ≤ ( n − n i )deg b i ( n − n i )+ m ( x ) ≤ pm + ( p + 1)( n − n i ) for ≤ m ≤ n i . Hence this requires n X k =1 (deg a k ( x ) − deg b ik ( x )) = n i ( n i + 1)2 linear equations of C ( ξ ) . Moreover P ( x, ∂ + α i x + β ij ) has the standard form P ( x, ∂ + α i x + β ij ) = n X k =0 p Y l =1 ( x − c l ) n − k c ijk ( x ) ∂ n − k , where deg c ijk ( x ) ≤ ( p + 1) k for ≤ k ≤ ( n − n i )deg c ij ( n − n i )+ l ( x ) ≤ pl + ( p + 1)( n − n i ) for ≤ l ≤ ( n i − n ij )deg c ij ( n − ( n i − n ij ))+ m ≤ ( p − m + p ( n i − n ij ) + ( p + 1)( n − n i ) for ≤ m ≤ n ij . Hence this requires n X k =1 (deg b ik ( x ) − deg c ijk ( x )) = n ij ( n ij + 1)2 linear equations of C ( ξ ) . Finally, informations about the sets of exponentsrequire t ij X k =1 n ijk ( n ijk + 1)2 linear equations of C ( ξ ) by Proposition 3.7. There is one more linear equa-tion from Fuchs relation. Hence there are at most the following parametersin P ( x, ∂ ) , n X i =1 (( p + 1) i + 1) − p X i =1 s i X j =0 m ij ( m ij + 1)2 − r X i =1 n i ( n i + 1)2 − r X i =1 l i X j =1 n ij ( n ij + 1)2 − p X i =1 l i X j =1 t ij X k =1 n ijk ( n ijk + 1)2 + 1 = 1 −
12 idx P. Theorem 4.18.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12 withthe table of local datum, α · · · α r c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] ... ... ... ... ... ... ... ... c p [ µ p ; m p ] β l [ ν l ; n l ] β l [ ν l ; n l ] β rl r [ ν rl r ; n rl r ] .1. For f ( ξ ) ∈ C ( ξ ) and i = 1 , . . . , p , the table of local datum of RAd(( x − c i ) f ( ξ ) ) P ( x, ∂ ) is α · · · α r c [ µ ; m ] β [ ν − f ( ξ ); n ] · · · β r [ ν r − f ( ξ ); n r ] c [ µ ; m ] β [ ν − f ( ξ ); n ] · · · β r [ ν r − f ( ξ ); n r ] ... ... ... ... ... ... c i [ µ i + f ( ξ ); m i ] ... ... ... ...... ... ... ... ... ... c p [ µ p ; m p ] β l [ ν l − f ( ξ ); n l ] β rl r [ ν rl r − f ( ξ ); n rl r ] .where µ i + f ( ξ ) = ( µ i + f ( ξ ) , . . . , µ is i + f ( ξ )) and ν ij − f ( ξ ) = ( ν ij − f ( ξ ) , . . . , ν ijt ij − f ( ξ )) .2. For i = 1 , . . . , r , j = 1 , . . . , l i and k = 1 , . . . , t ij , the table of localdatum of E ( α i x + β ij x ; β ij , ν ijk − P ( x, ∂ ) is α α · · · α r c [˜ µ ; m ] β [˜ ν ; ˜ n ] β [˜ ν ; ˜ n ] · · · β r [˜ ν r ; ˜ n r ] c [˜ µ ; m ] β [˜ ν ; ˜ n ] β [˜ ν ; ˜ n ] · · · β r [˜ ν r ; ˜ n r ] ... ... ... ... ... ... ... ... c p [˜ µ p ; m p ] β l [˜ ν l ; ˜ n l ] β l [˜ ν l ; ˜ n l ] β rl r [˜ ν rl r ; ˜ n rl r ] ,where ˜ ν xy = ( ν ij − ( ν ijk − , . . . , ν ijk − − ( ν ijk − , − ν ijk , ν ijk +1 − ( ν ijk − , . . . ) if ( x, y ) = ( i, j ) ,ν iy if x = i and y = j, ( ν xy + ν ijk − , . . . , ν xyt xy + ν ijk − otherwise , ˜ µ x = ( µ x + ν ijk − , . . . , µ xs x + ν ijk − , ˜ n xy = ( ( n ij , . . . , n ijk − , N i − n ij , n ijk +1 , . . . ) if ( x, y ) = ( i, j ) ,n xy otherwise , or N i = P pk =1 P s k l =1 m kl +( n − n i ) . The order of E ( α i x + β ij x ; β ij , ν ijk − P ( x, ∂ ) is n − n ijk + N i − n ij .Proof. The first assertion is the direct consequence of Proposition 3.14 andProposition 3.10. The second assertion follows from Theorem 4.14 and thefirst assertion.
Remark 4.19.
In the second assertion of Theorem 4.18, we see that theorder of E ( α i x + β ij x ; β ij , ν ijk − P ( x, ∂ ) is ( n − n ijk ) + ( N − n ij ) . By thesame argument as the proof of Proposition 4.3, we can see that N i − n ij ≥ for all i and j . Hence E ( α i x + β ij x ; β ij , ν ijk − P ( x, ∂ ) is well-defined, i.e., ( n − n ijk ) + ( N i − n ij ) > , if and only if n − n ijk > or N i − n ij > . On thecontrary, if we assume n = n ijk and N i = n ij , we can see the following. Lemma 4.20.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] whose table of local datum is αc [ µ ; m ] β [(0); ( n )] c [ µ ; m ] ... ... c p [ µ p ; m p ] .And we assume n = P pi =1 P s i j =1 m ij . Then P ( x, ∂ ) ∼ ( ∂ − αx − β ) n . Proof.
We can see that x = ∞ is a regular singular point of P ( x, ∂ + αx + β ).Hence we can write P ( x, ∂ + αx + β ) = n X i =0 x n − i P i ( ∂ ) , where deg P i ( x ) ≤ n for i = 0 , . . . , n and P ( j ) i (0) = 0 for i + j < n . Weconsider polynomials p k ( λ, x ) = k X i =0 ( λ + i + 1) k − i P ( k − i ) i ( x )( k − i )! . The characteristic exponents at x = ∞ implies that p n + j ( − i − j,
0) = 0for i = 0 , , . . . , n − j = 0 , . . . , n − i − . This implies that P ( j ) i (0) = 0for i = 1 , . . . , n and j = 0 , , . . . n −
1. Hence we have P ( x, ∂ + αx + β ) ∼ ∂ n , P ( x, ∂ ) ∼ ( ∂ − αx − β ) n . Corollary 4.21.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12 withthe table of local datum, αc [ µ ; m ] β [( ν ); ( n )] c [ µ ; m ] ... ... c p [ µ p ; m p ] .And we assume that n = P ri =1 P s i j =1 m ij . Then we have E ( α x ; β, ν ) P ∼ ( ∂ − αx − β ) n Proof.
By Theorem 4.18, we can see that E ( α x ; β, ν ) P has the table oflocal datum αc [ µ ; m ] β [(0); ( n )] c [ µ ; m ]... ... c p [ µ p ; m p ] .Therefore by Lemma 4.20, we have E ( α x ; β, ν ) P ∼ ( ∂ − αx − β ) n . Proposition 4.22.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12 andassume that P ( x, ∂ ) has the nontrivial table of local datum α α · · · α r c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] ... ... ... ... ... ... ... ... c p [ µ p ; m p ] β l [ ν l ; n l ] β l [ ν l ; n l ] β rl r [ ν rl r ; n rl r ] .Then we have the followings.1. We have E ( α i x ; ∗ , P ∼ P. for i = 1 , . . . , r and for any complex number ∗ . . Let us take i ∈ { , . . . , r } and j ∈ { , . . . , l i } and fix them. If ν ijk − f ( ξ ) / ∈ Z \{ } for all k = 1 , . . . , t ij , then we have E ( α i x ; β ij ; − f ( ξ )) E ( α i x ; β ij ; f ( ξ )) P ∼ P. Proof.
By the assumption, we can writeR P ( x, ∂ + α i x ) = x N l i Y j =1 ( ∂ − β ij ) n ij + N X k =1 x N − k l i Y j =1 ( ∂ − β ij ) max { n ij − k } P k ( ∂ )for P k ( x ) ∈ C [ x ] and N = n − n i + P pk =1 P s k l =1 m kl . Hence we can applyProposition 4.9.Although the twisted Euler transform in Corollary 4.21 does not satisfythe assumption of 2 in Proposition 4.22, we can show the following. Proposition 4.23.
Let us take P ( x, ∂ ) as in Definition 4.12 with the tableof local datum, αc [ µ ; m ] β [( ν ); ( n )] c [ µ ; m ] ... ... c p [ µ p ; m p ] .Then we have E ( α x ; β, − ν ) E ( α x ; β, ν ) P ∼ P. Proof.
Without loss of the generality, we can assume α = 0. The table oflocal datum of RAd(( x + β ) − ν ) L R P is 0 − β [(0 , − N, n )] c [ µ ′ ; m ] c [ µ ′ ; m ]... ... c p [ µ ′ p ; m p ] .Here N = P ri =1 P s i j =1 m ij − n and µ ′ i = ( µ i + ν + 1 , . . . , µ is i + ν + 1)for i = 1 , . . . , r . Hence by the same argument as in Proposition 4.9, we canshow that( L − R L )RAd(( x + β ) − ν ) L − P ∼ RAd(( x + β ) − ν ) L − P. E ( β, − ν ) E ( β, ν ) P = L RAd(( x + β ) ν ) L − R L RAd(( x + β ) − ν ) L − R P = L RAd(( x + β ) ν )RAd(( x + β ) − ν ) L − P = L R L − P = E ( ∗ , P ∼ P. P. Boalch found a correspondence between quiver varieties and moduli spacesof meromorphic connections on vector bundles over the Riemann sphere ofthe forms ( Az + Bz + Cz ) dz. He studied the existence of these meromorphic connections through thetheory of representations of quiver varieties which is first studied by W.Crawley-Boevey in [4].In this section, as an analogue of this result of Boalch, we attach adifferential operator considered in Definition 4.12 to a Kac-Moody Lie al-gebra and an element of the root lattice of this algebra. And we show theequivalence between twisted Euler transforms and additions on differentialequations and the action of Weyl group on the corresponding element of theroot lattice.Let us take P ( x, ∂ ) ∈ W [ x, ξ ] with the table of local datum α α · · · α r c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ] c [ µ ; m ] β [ ν ; n ] β [ ν ; n ] · · · β r [ ν r ; n r ]... ... ... ... ... ... ... ... c p [ µ p ; m p ] β l [ ν l ; n l ] β l [ ν l ; n l ] β rl r [ ν rl r ; n rl r ] .We fix this P ( x, ∂ ) through this section.Let h be the complex vector space with the basisΠ = { v ijk | i = 0 , . . . , r, j = 1 , . . . , l i , k = 1 , . . . , t ij } . Here we put l = p and t j = s j . We define the non-degenerate symmetricbilinear form on h as follows,( v ijk , v lmn ) = i, j, k ) = ( l, m, n ) − i, j ) = ( l, m ) and | k − n | = 1 − k, n ) = (1 ,
1) and i = j .
38f for s = ( i s , j s , k s ) ∈ I = { ( i, j, k ) | i = 0 , . . . , r, j = 1 , . . . , l i , k =1 , . . . , t ij } , we write v s = v i s j s k s . We can define the generalized Cartan matrix, A = (cid:18) v s , v t )( v s , v s ) (cid:19) s ∈ I,t ∈ I . Let g ( A ) be the Kac-Moody Lie algebra with the above generalized Cartanmatrix A . According to the usual terminology, we call Π the root basis,elements from Π are called simple roots and Z -lattice generated by Π, i.e. Q = X ( i,j,k ) ∈ I Z v ijk is called the root lattice. Also we define the positive root lattice Q + = X ( i,j,k ) ∈ I Z ≥ v ijk . The height of an element of the root lattice α = P i,j,k ∈ I x ijk v ijk is defined byht( α ) = X ( i,j,k ) ∈ I x ijk . Also the support of α is defined bysupp α = { v ijk ∈ Π | x ijk = 0 } . We say that the subset L ⊂ Π is connected if the decomposition L ∪ L = L with L = ∅ and L = ∅ always implies the existence of v i ∈ L i satisfying( v , v ) = 0.We have the following root space decomposition of g ( A ) with respect to h , g ( A ) = M α ∈ Q g α where g α = { X ∈ g ( A ) | [ H, X ] = ( α, H ) X for all H ∈ h } is the root spaceattached to α . The root space is ∆ = { α ∈ Q | g α = { }} and we callelements of ∆ roots.The reflections on h with respect to simple roots v ijk , so-called simplereflections, are defined by r ijk : h ∋ H r ijk ( H ) = H − H, v ijk )( v ijk , v ijk ) v ijk = H − ( H, v ijk ) v ijk . The Weyl group W is the group generated by all simple reflections.A root α ∈ ∆ is called real root if there exists w ∈ W such that w ( α ) ∈ Π.We denote the set of real roots by ∆ re . A root which is not real root is called39maginary root. We denote the set of all imaginary roots by ∆ im . Hencethere is a decomposition of the set of roots, ∆ = ∆ re ∪ ∆ im . If the Cartanmatrix is symmetrizable, we can see that∆ re = { α ∈ ∆ | ( α, α ) > } , ∆ im = { α ∈ ∆ | ( α, α ) ≤ } . In our case the Cartan matrix A is symmetric. Hence moreover we have∆ re = { α ∈ ∆ | ( α, α ) = 2 } . For fundamental things about Kac-Moody Lie algebra, we refer the stan-dard text book [7].For the above P ∈ W [ x, ξ ], we can define the element α P ∈ Q + associ-ated with P ( x, ∂ ) as follows. Let us put˜ n ijk = (P s j l = k m jl if i = 0 P t ij l = k n ijl if i = 1 , . . . , r . Then α P ∈ Q + is defined by α P = X ( i,j,k ) ∈ I ˜ n ijk v ijk . Remark 5.1.
This correspondence between P ∈ W [ x, ξ ] and α P ∈ Q + is not unique. Indeed, for ν ijk and n ijk , permutations with respect to theindex k = 1 , . . . , t ij do not change local solutions of P . For µ ij and m ij ,permutations with respect to the index j = 1 , . . . , s i do not change localsolutions of P as well. Example 5.2. If P ( x, ∂ ) has the table of local datum, α α c [( µ ); (1)] β [( ν ); (1)] β [( ν ); (1)] ,the corresponding Kac-Moody Lie algebra has the following dynkin diagram,.If P ( x, ∂ ) has the table of local datum, αc [( µ , µ ); (1 , β [( ν , ν ); (1 , c [( µ , µ ); (1 , β [( ν , ν ); (1 , β [( ν , ν ); (1 , , he corresponding Kac-Moody Lie algebra has the following dynkin diagram,. Theorem 5.3.
We retain the above notations.1. For P ∈ W [ x, ξ ] and α P ∈ Q + defined as above, we have idx P = ( α P , α P ) .
2. Let us assume n − n ijk > or N i − n ij > where N i = P l k =1 P t k l =1 m kl +( n − n i ) . If we put Q = E ( α i x + β ij x ; β ij , ν ij − P ( x, ∂ ) , then we have α Q = r ij ( α P ) .
3. If we put Q = RAd(( x − c i ) − µ i ) P , then we have α Q = r i ( α P ) .
4. For k ≥ , reflections r ijk ( α P ) correspond following permutations ofthe table of local datum of P , ( µ yz , m yz ) ( µ yz +1 , m yz +1 ) if i = 0 , j = y and k = z ( µ yz +1 , m yz − ) if i = 0 , j = y and k = z − µ yz , m yz ) otherwise , ( ν xyz , n xyz ) ( ν xyz +1 , n xyz +1 ) if ( x, y, z ) = ( i, j, k )( ν xyz − , n xyz − ) if ( x, y, z −
1) = ( i, j, k )( ν xyz , n xyz ) otherwise . Here µ yz and ν xyz are exponents of local solutions of P and m yz and n xyz are multiplicities of them respectively. This theorem immediately follows from the following lemma.
Lemma 5.4.
Let us take α ∈ h such that α = X ( i,j,k ) ∈ I c ijk v ijk or c ijk ∈ C . The we have the following equations, ( α, α ) = − ( l + 1)( r X i =1 l i X j =1 c ij ) + r X i =1 ( l i X j =1 c ij ) + r X i =1 l i X j =1 ( c ij ) + l X i =1 ( r X j =1 l j X k =1 c jk − c i ) + r X i =0 l i X j =1 t ij X k =1 ( c ijk − c ijk +1 ) , ( α, v i j k ) = − ( r X i =1 l i X j =1 t ij X k =1 c ijk − l i X i =1 t i i X j =1 c i jk ) + c i j + ( c i j − c i j ) if k = 1( c i j k − c i j k − ) + ( c i j k − c i j k +1 ) if k ≥ . Proof.
Direct computation.We consider a subset of Q + , V = { nv i ′ j ′ + l X j =1 n X k =1 m jk v jk | ≤ i ′ ≤ r, ≤ j ′ ≤ l i ′ n,m jk ∈ Z > such that n ≥ m j >m j > ··· } Proposition 5.5. If α P / ∈ W ( V ) , the set of all Weyl group orbits of ele-ments of V , then the Weyl group orbit of α P , W ( α P ) , is contained in Q + .Proof. If α P / ∈ V , a twisted Euler transform of P ( x, ∂ ) corresponds to asimple reflection of α P by Theorem 5.3. Hence we can find Q ( x, ∂ ) ∈ W [ x, ξ ]which corresponds to α Q = r ij ( α P ) by taking the twisted Euler transformof P ( x, ∂ ). Moreover if α Q / ∈ V , we can find Q ′ ( x, ∂ ) ∈ W [ x, ξ ] whichcorresponds to α Q ′ = r i ′ j ′ ( α Q ) = r i ′ j ′ r ij ( α P ) by the twisted Euler transform.Hence if α P / ∈ W ( V ), we can iterate these. Also we can use same argumentfor other simple reflections. Then we have the proposition. Corollary 5.6. If α P / ∈ W ( V ) , then α P ∈ ∆ im .Proof. First we assume that ( α P , α P ) >
0. Let β be an element of minimalheight among W ( α P ) ∩ Q + . Since the Weyl group action does not changeinner product, we have ( β, β ) >
0. Hence we have ( β, v ijk ) > i, j, j ) ∈ I . If β = v ijk , then r ijk ( β ) ∈ Q + and ht( r ijk ( β )) < ht( β ), a con-tradiction with the choice of β . Hence β = v ijk . Since α P / ∈ W ( V ), we canfind Q β ∈ W [ x, ξ ] such that α Q β = β . This implies that if β is the simpleroot, then β = v ij for some i and j . However v ij ∈ V . This contradicts ourassumption. Hence if α P / ∈ W ( V ), we have ( α P , α P ) ≤ α P , α P ) ≤
0. As above, we choose an element β ∈ W ( α P ) of minimal height. Then ( β, v ijk ) ≤ i, j, k ) ∈ I . Since42 corresponds to some Q β ∈ W [ x, ξ ], this implies that supp β is connected.Hence β ∈ K = { α ∈ Q + \{ } | ( α,v ijk ) ≤ i, j, k ) ∈ I and supp α is connected } . This implies that α P ∈ W ( K ) = ∆ im ∩ Q + . Theorem 5.7.
Let us take P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12. If idx P > , then P ( x, ∂ ) can be reduced to ( ∂ − αx − β ) n for some α, β ∈ C and n ∈ Z > by finite iterations of twisted Euler trans-forms and additions at regular singular points.Proof. By Corollary 5.6, if idx P = ( α P , α P ) >
0, then α P ∈ W ( V ). Hencefinite iterations of simple reflections α P reduces to an element of V . Thisimplies P reduces to a Q ∈ W [ x, ξ ] with a table of local datum αc [ µ ; m ] β [( ν ); ( n )] c [ µ ; m ]... ... c p [ µ p ; m p ] ,by finite iterations of twisted Euler transforms and additions at regularsingular points. Proposition 4.21 says that E ( α x ; β, ν ) Q ∼ ( ∂ − αx − β ) n . Hence we have the theorem.
In this section, we show that differential operator P ( x, ∂ ) of idx P >
Definition 6.1.
Let us take P ( x, ξ ) ∈ W [ x, ξ ] as in Definition 4.12. If P ( x, ∂ ) has the following table of local datum, c [ µ ; m ] 0 [( ν , . . . , ν t ); ( n , . . . , n t )] c [ µ ; m ] ... ... c p [ µ p ; m p ] , hen we say that P ( x, ∂ ) is Fuchsian. Proposition 6.2. If P ( x, ∂ ) is Fuchsian with nontrivial table of local datum,then E (0 , f ( ξ )) P and RAd(( x − c ) g ( ξ ) ) P are Fuchsian for any c ∈ C and f ( ξ ) , g ( ξ ) ∈ C ( ξ ) .Proof. This is a collorary of Theorem 4.18.
We define the operator called versal additions. These operators are intro-duced by Oshima in [11].For a , . . . , a n ∈ C , we define a function h n ( a , . . . , c n ; x ) = − Z x t n − dt Q ≤ i ≤ n (1 − a i t ) . Then it is not hard to see that e λ n h n ( a ,...,a n ; x ) = n Y k =1 (1 − a k x ) λnak Q ≤ i ≤ ni = k ( ak − ai ) . Definition 6.3 (Versal addition) . We put
AdV( a , . . . , a n ; λ , . . . , λ n ) = n Y k =1 Ad ( x − a k ) P nl = k λlak Q ≤ i ≤ li = k ( ak − ai ) . Proposition 6.4.
For P ( x, ∂ ) ∈ W [ x ] , we have lim a → AdV( a ; λ ) P ( x, ∂ ) = Ade( − λ x ) P ( x, ∂ ) , lim a → a → AdV( a , a ; λ , λ ) P ( x, ∂ ) = Ade( λ x + λ x ) P ( x, ∂ ) . Proof.
If we recall thatAdV( a ; λ ) : ∂ = ∂ − λ a ( x − a ) = ∂ + λ − a x , then we can see thatlim a → AdV ( a ; λ ) ∂ = ∂ + λ = Ade( − λ x ) ∂. a , a ; λ , λ ) ∂ = ∂ − λ a + λ a ( a − a ) x − a − λ a ( a − a ) x − a = ∂ + λ + λ ( a − a ) − a x + λ ( a − a ) − a x = ∂ + λ − a x + λ x (1 − a x )(1 − a x ) , implies thatlim a → a → AdV( a , a ; λ , λ ) ∂ = ∂ + λ + λ x = Ade( − λ x − λ x ) ∂. Theorem 6.5.
Take a P ( x, ∂ ) ∈ W [ x, ξ ] as in Definition 4.12. If idx P > ,then P ( x, ∂ ) can be obtained by the limit transition of a Fuchsian Q ( x, ∂ ) ∈ W [ x, ξ ] of idx Q = idx P. Proof.
By Theorem 5.7, P ( x, ∂ ) is obtained by finite iterations of twistedEuler transforms and additions from ( ∂ − αx − β ) n . Here ∂ − αx − β =lim a → a → AdV( a , a ; − α, − β ) ∂. As we see in Remark 2.11, twisted Eulertransforms are compositions of Euler transforms, Ade( αx ) and Ade( βx + γx ) for some α, β, γ ∈ C . Hence twisted Euler transforms can be obtainedby the limit transitions of compositions of additions and Euler transformsby Proposition 6.4.Therefore P ( x, ∂ ) can be seen as the limit of a Fuchsian Q ( x, ξ ) which isobtained by Euler transforms and additions from AdV( a , a ; − α, − β ) ∂. Finally we notice that twisted Euler transforms do not change rigidityindices because the action of Weyl group does not change the inner product.
AppendixA Differential equations with regular singularityat x = ∞ and Euler transform We consider differential equations with regular singular point at x = ∞ and arbitrary singularities at any other points in C . And then we give anecessary and sufficient condition to reduce the rank of differential equationby Euler transform. 45 heorem A.1. Let us take P ( x, ∂ ) ∈ W [ x ] which has regular singular pointat x = ∞ and semi-simple exponents { [ µ ] n , . . . , [ µ l ] n l } , where P li =1 n i = n = ord P , µ i / ∈ Z and µ i − µ j / ∈ Z if i = j . Then we have ord E (0 , µ i − P ( x, ∂ ) < ord P if and only if deg P − ord P < n i . Proof.
Since x = ∞ is the regular singular point of P ( x, ∂ ) ∈ W [ x ], we canwrite R P ( x, ∂ ) = N X i =0 x N − i ∂ max { n − i, } P i ( ∂ )for P i ( x ) ∈ C [ x ] of deg P i ≤ n for i = 0 , . . . , N and P ( x ) = 1. Here N = deg P and n = ord P . Hence we have L − R P = N X i =0 ∂ N − i ( − x ) max { n − i } P i ( − x )and this has regular singular point at x = 0 and no other singular points in C . Also this has semi-simple exponents, { [0] N − n , [ µ − n , . . . , [ µ l − n l } at x = 0 by Proposition laplace inverse transform of regular point. Andthen we can see that RAd( x − µ i +1 ) L − R P has semi-simple exponents, { [ − µ i + 1] N − n , [ µ − µ i ] n , . . . , [ µ i − − µ i ] n i − , [0] n i [ µ i +1 − µ i ] n i , . . . } by Proposition 3.14. Hence by Proposition 3.12, we havedeg RAd( x − µ i +1 ) L − R P = n − n i + ( N − n ) . This means thatord L RAd( x − µ i +1 ) L − R P = E (0 , µ i − P = n − n i + ( N − n ) . Hence we have the theorem. 46 eferences [1] Bertrand, D.: On Andr´e’s proof of the Siegel-Shidlovsky theorem. Col-loque Franco-Japonais: Th´eorie des Nombres Transcendants (Tokyo,1998), 51–63, Sem. Math. Sci., , Keio Univ., 1999.[2] Bertrand, D. and Laumon, G.: Appendix of Exposants dessyst´emes diff´erentiels, vecteurs cycliques et majorations de multi-plicit´es. ´Equations diff´erentielles dans le champ complexe, Vol. I (Stras-bourg, 1985), 61–85, Publ. Inst. Rech. Math. Av., Univ. Louis Pasteur,Strasbourg, 1988.[3] Boalch, P.: Irregular connections and Kac-Moody root systems.preprint, arXiv:0806.1050.[4] Crawley-Boevey, W.: On matrices in prescribed conjugacy classes withno common invariant subspace and sum zero. Duke Math. J. (2003),no. 2, 339–352.[5] Dettweiler, M. and Reiter, S.: An algorithm of Katz and its applicationto the inverse Galois problem. J. Symbolic Comput. (2000), no. 6,761-798.[6] Dettweiler, M. and Reiter, S.: Middle convolution of Fuchsian systemsand the construction of rigid differential systems. J. Algebra (2007),no. 1, 1-24.[7] Kac, V.: Infinite dimensional Lie algebras, Third edition. CambridgeUniv. Press 1990.[8] Katz, N.: Rigid local systems. Annals of Mathematics Studies, .Princeton University Press, Princeton, 1996. viii+223 pp.[9] Kawakami, T.: Generalized Okubo systems and the middle convolution,Thesis, The University of Tokyo, 2009.[10] Malgrange, B.: Sur la r´eduction formelle des ´equations diff´erentielles´a singularit´es irr´eguli´eres. Singularit´es irr´eguli´eres, Correspondance etdocuments, Documents Math´ematiques, . Soci´et´e Math´ematique deFrance, Paris, 2007. xii+188 pp.[11] Oshima, T.: Fractional calculus of Weyl algebra and Fuchsian differen-tial equations. preprint.[12] Ramis, J.-P.: Devissage Gevrey. Ast´erisque,59-60