Fuchs Fuchs Laplace Katz [Kz] middle convolution addition Gauss Airy Fuchs addition middle convolution Fuchs 5 Fuchs Riemann, rigidity

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2 Fuchs Fuchs Laplace Katz [Kz] middle convolution addition Gauss Airy Fuchs addition middle convolution Fuchs 5 Fuchs Riemann, rigidity

3 0 Fuchs addition middle convolution Riemann Fuchs rigidity 2 rigid Fuchs Katz Fuchs 7 Riemann Fuchs Riemann Fuchs Kac-Moody rigidity Riemann Fuchs rigid non-rigid Weyl Fuchs rigid [O6] trivial rigid Jordan-Pochhammer even/odd 3 rigid 2010 Tsai [Hi2]

4 (Fractional operators Weyl versal addition versal operator Fuchs Riemann Fuchs Fuchs Riemann Katz Riemann

5 Fuchs Kac-Moody Weyl Deligne-Simpson-Katz Gauss Jordan-Pochhammer P n H n even/odd EO n

6 3 1 Gauss Bessel Whittaker Legendre *1 Gauss III [MUI] 3 2 Gauss *2 Gauss Euler F (α, β, γ; x = i=0 (α n (β n x n = 1 + αβ (γ n n! γ x +. Γ(α + n α, β, γ C *3 (α n := α(α + 1 (α + n 1 = Γ(α x < 1 Gauss x(1 xu + ( γ (α + β + 1x u αβu = 0. P = x(1 x 2 + ( γ (α + β + 1x αβ ( := d dx Gauss (GE: P u = 0 : x λ λx λ 1 *1 Whittaker-Watson [WW] Watson [Wa] *2 *3 γ

7 4 1 x: x λ x λ+1 ϑ := x : x λ λx λ x λ ϑ λ xp = x γx x(x (α + β + 1x + αβ = ϑ(ϑ 1 + γϑ x(ϑ(ϑ 1 + (α + β + 1ϑ + αβ = ϑ(ϑ + γ 1 x(ϑ + α(ϑ + β. P u = 0 ϑ(ϑ + γ 1u = x(ϑ + α(ϑ + βu. u u(x = n=0 c nx n x n c n n(n + γ 1 = c n 1 (n 1 + α(n 1 + β c n = c n 1 (α + n 1(β + n 1 (γ + n 1n = = c 0 (α n (β n (γ n n! ( c0 = u(0 Gauss P u = 0 x = 1 Gauss C α,β,γ := F (α, β, γ; 1 = Γ(γΓ(γ α β Γ(γ αγ(γ β Re α + Re β < Re γ *4 Gauss (Gauss summation formula Gauss 2 1 Gauss C α,β,γ+1 C α,β,γ = γ(γ α β (γ α(γ β lim n C α,β,γ+n = lim n C α,β,γ+n = 1 Gauss 2 *5 F (α, β, γ; x F (α, β, γ; x = Γ(γ Γ(βΓ(γ β 1 0 t β 1 (1 t γ β 1 (1 tx α dt x = 1 Re (γ β α > 0 ( = Γ(γ Γ(βΓ(γ α β Γ(βΓ(γ β Γ(γ α *4 Re α α Im α *5 0 < Re β < Re γ, 0 < arg(x 1 < 2π

8 5 (1 tx α = = = n=0 n=0 n=0 ( α ( t n x n n ( α( α 1 ( α n + 1 ( t n x n n! (α n t n x n n! x n Γ(γ (α n Γ(β + nγ(γ β = (α n(β n Γ(βΓ(γ β n! Γ(γ + n (γ n n! F (α, β, γ; x (GE *6 {0, 1, } U (C { }\{0, 1, } F(U = {u O(U P u = 0} U O(U U dim C F(U = 2. U U F(U U = F(U. F(U = Cφ 1 + Cφ 2 φ 1, φ 2 O(U 1. C R >0, N N φ j (x C x N (j = 1, 2, x U, arg x < 2π, x < 1 2 x = 0 *7 2. x = 0 ψ 1 (0 = ψ 2 (0 = 1 ψ 1 (x, ψ 2 (x F(U = Cx 0 ψ 1 (x + Cx 1 γ ψ 2 (x x = 0 (characteristic exponent 0 1 γ ψ 1 (x = F (α, β, γ; x x = 1 x = {0, γ α β} {α, β} x = α 1 γ γ α β β Riemann (Riemann scheme Riemann Gauss Gauss *8 *6 1 x 1 x *7 *8 rigid

9 6 1 (monodromy (C { }\{0, 1, } x 0 γ 0, γ 1, γ γ 0 x γ 1 (u 1, u 2 x 0 (GE γ j (j = 0, 1, x 0 γ j (u 1, u 2 M j GL(2, C γ j (u 1, u 2 = (u 1, u 2 M j γ M j (j = 0, 1, x = j ( u (GE x µ 1 (1 x µ 2 u Riemann x = 0 1 µ 1 µ 2 α µ 1 µ 2 µ γ µ 2 + γ α β β µ 1 µ 1 Riemann x = 0 1 λ 01 λ 11 λ 21 λ 02 λ 12 λ 22 λ kl (Fuchs λ kl = 1 Riemann u 02 x = 0 λ 02 u 12 x = 1 λ 12 u 01 u 22 a, b ( ( e(λ02 a e(λ11 0 M 0 =, M 0 e(λ 01 1 = b e(λ 12 *9 e(λ := e 2πiλ M = (M 1 M 0 1 Tr(M 1 M 0 = Tr(M 1 * 10 e(λ 02 e(λ 11 + e(λ 01 e(λ 12 + ab = e( λ 21 e( λ 22 *9 u 12 = u 01 + Cu 02 γ 0 u 12 = e(λ 01 u 01 + Ce(λ 02 u 02 = e(λ 01 u 12 + C ( e(λ 02 e(λ 01 u 02 M 1 λ kl *10 γ 0, γ 1, γ M M 1 M 0

10 7 ab 0 * 11 a = 1 b a, b Riemann rigid n P u = 0 P * 12 {c 0 =, c 1,..., c p } ( C { } x = c j ( nj m j,1,..., m j,nj ν=1 m j,ν = n n m j = (m j,1,..., m j,nj m = (m 0, m 1,..., m p n (p + 1 Riemann (generalized Riemann scheme x = c 0 c 1 c p [λ 0,0 ] (m0,0 [λ 1,0 ] (m1,0 [λ p,0 ] (mp,0... [λ 0,n0 ] (m0,n0 [λ 1,n1 ] (m1,n1 [λ p,np ] (mp,n p [λ] (k := λ λ + 1. λ + k 1 λ j,ν λ j,ν 0 j p, 1 ν < ν n j * 13 Riemann Riemann P m (λu = 0 Riemann Deligne-Simpson-Katz *11 *12 *13 λ j,ν

11 8 1 n Π = {α 0, α 0,1, α 0,2,..., α 1,1, α 1,2,..., α j,ν,...} α 0,1 α 0,2 Dynkin n m n j,ν = n j µ=ν+1 m j,µ α 0 α 1,1 α 1,2 α 2,1 α 2,2 α 3,1 α 3,2 m α m = nα 0 + j 0 n j,ν α j,ν ν 1 α m Deligne-Simpson-Katz rigidity * 14 Riemann α m rigid α m n F n 1 Jordan- Pochhammer rigid 4 2 Heun rigid α s α Weyl Riemann {P m (λ} Weyl {P m (λ} + = {α m } S α s α {P m (λ} + = {α m } S α µ j (x c j λ j * 15 Riemann Fuchs Appell Painlevé *14 0 {m j,ν } 1 *15

12 9 2 (Fractional operators 2.1 Weyl 0 Weyl Weyl C[x] = C[x 1, x 2,..., x n ] x 1, x 2,..., x n C(x x 1, x 2,..., x n i = (i = 1, 2,..., n Weyl x i W [x] x 1, x 2,..., x n 1, 2,..., n C Leibniz [a, b] = ab ba Weyl [x i, x j ] = [ i, j ] = 0, [ i, x j ] = δ ij (1 i, j n. x x = x x, i ( xj u(x = (x j i + δ ij u(x W [x] C[x] W (x Weyl W (x = C(x C[x] W [x] f, g C[x] [ i, g f g ] = x i f g f x i f 2 (2.1 Weyl ξ 1, ξ 2,..., ξ m C[ξ], C(ξ ξ 1, ξ 2,..., ξ m Weyl

13 10 2 (Fractional operators W [x][ξ], W [x; ξ] Weyl W (x; ξ W [x][ξ] = C[ξ] C W [x], W [x; ξ] = C(ξ C W [x], W (x; ξ = C(ξ C W (x. [x i, ξ ν ] = [ i, ξ ν ] = 0 (1 i n, 1 ν m, C C[x; ξ] = C(ξ C C[x], C(x; ξ = C(ξ C C(x C C[x, ξ] W [x][ξ] W [x; ξ] W (x; ξ P W (x; ξ P = α=(α 1,...,α n N n p α (x, ξ α 1+ α n x α 1 1 xα n n ( pα (x, ξ C(x; ξ Proof. (2.1 P p α (x, ξ 0 P 0 p α (x, ξ 0 α := α α n > α p α (x, ξ = 0 α P x α = α 1! α n! p α (x, ξ 0 P 0 P (order ord P = max{ α p α (x, ξ 0} P W [x; ξ] (degree deg P = max{deg x p α (x, ξ} deg x p α (x, ξ x i (reduced representative W (x; ξ W [x; ξ] R P W (x; ξ R P W [x; ξ] (C(x; ξ\{0}p W [x; ξ] deg R P ii Fourier-Laplace I {1,..., n} L I : W [x; ξ] W [x; ξ] i x i x i i j j x j x j ξ ν ξ ν (i I, j / I, ν = 1,... m

14 2.1 Weyl 11 W [x, ξ] Fourier-Laplace L I I = {1,..., n} L = L {1,...,n} iii Fourier-Laplace W (x; ξ C(ξ C C[x] C C( C L: W (x; ξ C(ξ C C[x] C C( C(ξ C C[x] C C( P Q = L (L 1 P L 1 Q C W L (x; ξ i ii n = 1 ( 1 R x j 1 = x i x i ( R (x j ξ i = x i x i ( c m x = x( c m + m( c m 1 W [x; ξ] W L (x; ξ m x = x m m m 1 n = 1 P W (x P = m i=0 a i(x i ζ ζ P P (x, ζ = m a i (xζ i (Leibniz. P, Q W (x R = P Q W (x *1 R(x, ζ = ν=0 i=0 1 ν P (x, ζ ν Q(x, ζ ν! ζ ν x ν Proof. P = m i=0 a i(x i, Q = n i=0 b i(x i f R(x, ζ = P Qf = m = i=0 j=0 m i=0 j=0 m i=0 j=0 n a i (x n a i (x i (b j (x j f n a i (x i k=0 i ( i b k (k j (x i k+j f k=0 i(i 1 (i k + 1 k! b (k j (xζ i k+j *1 n > 1, ν ν = (ν 1,..., ν n ν! = ν 1 ν n, ν = ν 1 1 ν n n, ξ ν = ξ ν 1 1 ξν n n ν j

15 12 2 (Fractional operators = = m k=0 k=0 1 k! m a i (xi(i 1 (i k + 1ζ i k i=k 1 k P (x, ζ k Q(x, ζ k! ζ k x k n j=0 b (k j (xζ j ( (gauge transformation. h = (h 1,..., h n C(x, ξ n W (x; ξ Adei(h h j x i = h i x j (1 i, j n. Adei(h: W (x; ξ W (x; ξ i i h i x i x i ξ ν ξ ν [ i h i, j h j ] = h j x i + h i x j = 0 [ i h i, x j ] = δ ij g(x g x i = h i (i = 1,..., n f = e g Ad(f := Ade (g := Adei (h 1,..., h n = Adei (h n = 1 h = λ x g = λ log x, f = xλ Ad(x λ = λ x Leibniz Ad(x λ = x λ x λ (. φ = (φ 1,..., φ n C(x 1,..., x m, ξ n ( φj Φ = x i 1 i m 1 j n (m n n Φ Ψ = ( ψ i,j (x, ξ 1 i n ΨΦ n 1 j m m = n Ψ T φ : W (x 1,..., x n ; ξ W (x 1,..., x m ; ξ

16 2.1 Weyl 13 T φ(x i = φ i (x Tφ ( x i = m (1 i n, ψ i,j (x, ξ x j (1 i n x = y 2 x = 1 2y y, φ = x 2 [ 1 2x, x2 ] = 1 T φ f = e g, h i = g x i RAd (f = RAde (g = RAdei (h 1,..., h n := R Adei (h 1,..., h n, AdL (f = AdeL (g = AdeiL (h 1,..., h n := L Adei(h 1,..., h n L 1, RAdL (f = RAdeL (g = RAdeiL (h 1,..., h n := L RAdei (h 1,..., h n L 1, Ad( µ x i := L Ad(x µ i L 1, RAd ( µ x i := L RAd (x µ i L 1. µ C(ξ Ad( µ x i W L (x; ξ n = 1 Ad(f f f 1 = e g e g = e g (e g g dg e dx = h = Ad(f f P u(x = 0 ( Ad(fP (fu = 0 Ad(f u fu Ad(x µ = x µ x µ = x µ (x µ + µx µ 1 = + µx 1 RAd (x µ = x + µ Ad( µ Ad( µ x m = µ x m µ ( = = x m µ + ( µm x m 1 µ 1 + 1! ( µ( µ 1 ( µ m + 1m! + + m! m ( ( 1 ν m (µ ν x m ν ν ν ν=0 ( µ( µ 1m(m 1 x m 2 µ 2 2! µ m µ (2.2

17 14 2 (Fractional operators Leibniz Laplace Euler P 1 (C = C { } C u(x v(x = e xt u(t dt C v(x Laplace v(x = Lu(x *2 Lu(x = ( te xt u(t dt = L( xu(x. C Lu(x = 1 [ e xt u(t ] x + 1 C x C e xt u (t dt [ e xt u(t ] = 0 C C * 3 L( u(x = xlu(x W [x] Laplace x, x P u = 0 L (P (L u = 0 u(x I λ c (u(x = 1 Γ(λ x c u(t(x t λ 1 dt Euler (Riemann-Liouville *4 Re λ > 0 lim x c (x c 1 ɛ u(x = 0 ɛ *5 u(t c x c x c u(t(x tλ 1 dt λ λ = 0, 1, 2,... Γ(λ λ = 0, 1, 2,... 1 I λ c u(x λ = 0, 1, 2,... Ic λ I c µ u(x = Ic λ+µ u(x. Ic n u(x = dn dx n u(x *2 u(x C *3 C C *4 I λ c Pochhammer (x+,c+,x,c (Ĩµ c (u(x := u(z(x z µ 1 dz c *5 c = lim x c x 1+ɛ u(x = 0 x

18 p(x n ( Ic λ ( p(xu(x = p(xi λ λ c u(x + 1 Leibniz ( p (xic λ 1 λ u(x + + p (n (xi λ n n c u(x d ( u(t(x t λ 1 = u (t(x t λ 1 d ( u(t(x t λ 1 dt dx I λ c (u = I λ c ( u d ( u(t(x t λ = u (t(x t λ d ( u(t(x t λ 1 dt dx = xu (t(x t λ 1 tu (t(x t λ 1 λu(t(x t λ 1 [ u(t(x t λ ] t=x t=c = xiλ c ( u I λ c (ϑu λi λ c (u 0 I λ c (ϑu = (ϑ λi λ c (u RAd( λ Leibniz P W [x] P u = 0 ( RAd( λ P I λ c u = n = 1 h(x, ξ ξ x g(x, ξ, f(x, ξ g = h, f = e g λ C\{0} Adei (h = h = Ad(e g = e g e g, Ad(fx = x, AdL (f =, Ad(f = h(x, ξ, AdL (fx = x + h(, ξ, Ad ( (x c λ = Ade ( λ log(x c ( λ = Adei, x c Ad ( (x c λ x = x, Ad ( (x c λ = λ x c, RAdL ( (x c λ x = ( cx + λ = x cx λ, RAdL ( (x c λ =, RAdL ( (x c λ ϑ = x( c λ (ϑ := x, Ad ( µ ϑ = AdL ( x µ ϑ = ϑ µ, Ad ( ( e λ(x cm λ(x c m x = x, Ad e m m = λ(x c m 1,

19 16 2 (Fractional operators RAdL ( {x e λ(x cm + λ( c m 1 (m 1, m x = ( c 1 m x + λ (m 1, T (x c mx = (x cm, T (x c m = 1 m (x c1 m du dx = 0 RAd (x µ RAd ( ν (Gauss P λ1,λ 2, µ := RAd ( µ RAd ( x λ 1 (1 x λ 2 ( = RAd ( µ R λ 1 x + λ 2 1 x = RAd ( µ ( x(1 x λ 1 (1 x + λ 2 x = RAd ( µ ( (ϑ λ 1 x(ϑ λ 1 λ 2 = Ad( µ ( (ϑ + 1 λ 1 (ϑ + 1(ϑ λ 1 λ 2 = (ϑ + 1 λ 1 µ (ϑ + 1 µ(ϑ λ 1 λ 2 µ = (ϑ + γ (ϑ + β(ϑ + α = x(1 x 2 + ( γ (α + β + 1x αβ α = λ 1 λ 2 µ β = 1 µ γ = 1 λ 1 µ u(x = I µ ( 0 x λ 1 (1 x λ 2 = 1 x t λ 1 (1 t λ 2 (x t µ 1 dt Γ(µ = xλ 1+µ Γ(µ s λ 1 (1 s µ 1 (1 xs λ 2 ds = Γ(λ 1 + 1x λ 1+µ Γ(λ 1 + µ + 1 F ( λ 2, λ 1 + 1, λ 1 + µ + 1; x (t = xs Gauss Riemann { } x = 0 1 λ 1 λ 2 λ 1 λ 2 ; x x λ 1 (1 x λ 2 RAd ( µ Riemann x = 0 1 x = µ ; x = 0 0 β ; x λ 1 + µ λ 2 + µ λ 1 λ 2 µ 1 γ γ α β α

20 *6 Gauss *7 (Airy P m = L Ad ( e xm+1 m+1 = L ( x m = x ( m (m = 1, 2, 3,... d m u dx m ( 1m xu = 0 u(x = e zm+1 m+1 e zx dz = C C ( z m+1 exp m + 1 zx dz C π (2k+1π m+1 m+1 m = 2 k = 1, 2,..., m 0 0 C C (Jordan-Pochhammer c 1,..., c p C\{0} ( p P λ1,...,λ p,µ := RAd ( µ RAd (1 c j x λ j ( = RAd ( µ R + p = RAd ( µ ( p 0 (x + q(x c j λ j 1 c j x = µ+p 1( p 0 (x + q(x µ = p p k (x p k P λ1,...,λ p,µu = 0 Jordan-Pochhammer p 0 (x = p (1 c j x, q(x = p 0 (x ( µ + p 1 p k (x = k ( α := β Γ(α + 1 Γ(β + 1Γ(α β + 1 p k=0 c j λ j 1 c j x, p (k 0 (x + ( µ + p 1 k 1 (α, β C q (k 1 (x, * *7 λ 1 + µ } Ad (x λ1 µ Riemann } { x=0 1 λ 1 µ 0 λ λ 2 +µ λ 2 { x=0 1 Riemann 1 γ 0 β 0 γ α β α ( x β 1 (1 x α α = λ 2, β = λ 1 + 1, γ = λ 1 + µ + 1 x 1 γ I γ β 0

21 18 2 (Fractional operators u j (x = 1 x Γ(µ 1 c j p (1 c ν t λ ν (x t µ 1 dt (j = 0, 1, 2,..., p, c 0 = 0 ν=1 Riemann [0] (p 1 [0] (p 1 [1 µ] (p 1 ; x λ 1 + µ λ p + µ λ 1 λ p µ x = 1 c 1 1 c p 2.3 n Weyl n = 1 M: P u = 0 P W (x; ξ M P u P u = 0 W (x; ξ W (x; ξ W (x; ξ M P Q Qu M P = W (x; ξ/w (x; ξp M W (x; ξ M P W (x; ξ M P M O x0,ξ o (x, ξ (x 0, ξ 0 M O x0,ξ 0 Hom W (x;ξ (M, O x0,ξ 0 {v(x, ξ O x0,ξ 0 P v = 0} I I(u M W (x; ξ Hom W (x;ξ (M, O x0,ξ 0 W (x; ξ 1. W (x; ξ P Q = 0 P = 0 Q = 0 2. W (x; ξ Euclid P, Q W (x; ξ (P 0 R, S W (x; ξ (ord R < ord P Q = SP + R. (2.3 Proof. S, R W (x; ξ (ord R < ord P Q = S P + R 0 = (S S P + (R R S S ord ( (S S P + (R R ord P

22 P 0 S = S R = R P = a n n + + a 1 + a 0 (a n 0, Q = b m m + + b 1 + b 0 (b m 0 (2.4 ord Q < ord P S = 0, R = Q (2.3 ord Q ord P ord Q ord Q = 0 ord Q < ord Q Q W (x; ξ Q = Q b m a 1 n m n P ord Q < ord Q S R W (x; ξ (ord R < ord Q < ord Q Q = S P + R S = S + b m a 1 n m n Q = SP + R 3. W (x; ξ Proof. ord Euclid 4. (Euclid P, Q W (x; ξ\{0} P, Q U W (x; ξ P = AU, Q = BU P = A U, Q = B U U = CU (A, B, C, A, B, U W (x; ξ M, N W (x; ξ U = MP + NQ P, Q Euclid U, M, Q 5. W (x; ξ M P W (x; ξ M = W (x; ξ/w (x; ξp Proof. P, Q W (x; ξ R W (x; ξ W (x; ξ/w (x; ξp W (x; ξ/w (x; ξq = W (x; ξ/w (x; ξr. M u 1,..., u r I i = {P W (x; ξ P u i = 0} (i = 1,..., r I i P i W (x; ξ M W (x; ξu i = W (x; ξ/w (x; ξpi r W (x; ξ/w (x; ξp i M i=i

23 20 2 (Fractional operators R W (x; ξ r W (x; ξ/w (x; ξp i = W (x; ξ/w (x; ξr i=1 W (x; ξ/w (x; ξr M W (x; ξ W (x; ξ/w (x; ξr W (x; ξ M Q W (x; ξ W (x; ξ/w (x; ξq = M P, Q W (x; ξ (2.4 W (x; ξ/w (x; ξp, W (x; ξ/w (x; ξq u, v *8 u, u,... n 1 u C(x; ξ v, v,..., m 1 v P, Q x 0 w = u + (x x 0 n v x 0 w, w,..., m+n 1 w *9 dim C(x;ξ W (x; ξw m + n = dim C(x;ξ W (x; ξ/w (x; ξp + dim C(x;ξ W (x; ξ/w (x; ξq W (x; ξw W (x; ξ/w (x; ξp W (x; ξ/w (x; ξq P W (x; ξ, P 0 1. W (x; ξ/w (x; ξp W (x; ξ 2. W (x; ξp W (x; ξ 3. P = QR Q, R W (x; ξ ord Q ord R = 0 4. Q W (x; ξ Q / W (x; ξp M, N W (x; ξ MP + NQ = 1 5. S, T W (x; ξ ST W (x; ξp ord S < ord P S = 0 T W (x; ξp *8 W (x; ξ W (x; ξ/w (x; ξp 1 W (x; ξ u v *9 P u = 0 x 0 u(x, ( ν u(x 0 (ν = 0,..., n 1 Qv = 0 w = u + (x x 0 n v x 0 w(x ( ν w(x 0 (ν = 0,..., m + n 1

24 Addition RAd ( (1 cx µ u = 0 RAd : (1 cx + cµ ( (1 cx +cµ u = 0 (1 cx µ c, µ (c, µ (c, cµ = λ ((1 cx + λ c 0 ( + λ e λx Ade ( λx Ade ( λx = + λ (1 cx λ c (1 cx λ c = exp ( Ad ( (1 cx λ c = Adei ( λ 1 cx x 0 λ 1 cx dx λ, c (c, λ = (0, 0 Ad ( (1 cx λ c c=0 = Adei ( λ c=0 1 cx = Adei( λ = Ade ( λx = Ad (e λx 3.2 versal addition h(c, x c C c 1,..., c n C h n (c 1,..., c n ; x h n (c 1,..., c n ; x = 1 2π h(z, x n 1 z =R (z c j dz c 1..., c n {z C z < R} R h n (c 1,..., c n ; x c 1,..., c n {z C z < R} c i

25 h n (c 1,..., c n ; x h n (c 1,..., c n ; x = n k=1 h(c k, x (c k c i 1 i n i k Proof. c i c i 1 h n (c 1,..., c n ; x = 2π h(z, x n 1 z =R (z c j dz n h(c k, x = (c k c i. k=1 1 i n i k h(c, x = c 1 log(1 cx = x c 2 x2 c2 3 x3 h(c, x = x cx dx (1 cxh (c, x = 1 h n(c 1,..., c n ; x n 1 i n(1 c i x i k (1 c i x = i n n k=1 1 i n i k 1 i n i k (1 c i x k=1 (c k c i = xn 1 1 i n i k (c k c i Proof. f(c 1,..., c n, x f x n 1 c i f(cc 1,..., Cc n, C 1 x = C 1 n f(c 1,..., c n, x (C C \ {0} x i (c 1,..., c n (1 n + i i < n 1 0 f x n 1 c 1 = 0 1 h n (c 1,..., c n ; 0 = 0 x t n 1 h n (c 1,..., c n ; x = 0 1 i n (1 c it dt Ad ( e λ nh n (c 1,...,c n ;x ( ( (1 c i x = e λ nh n (c 1,...,c n ;x 1 i n e λ nh n (c 1,...,c n ;x = ( = 1 i n n k=1 1 i n (1 c i x + λ n x n 1, ( 1 c k x c k 1 i n i k (1 c i x e λ nh n (c 1,...,c n ;x λn (c k c i

26 3.3 versal operator (Versal addition. ( p ( AdV ( 1 c,..., 1 1 cp (λ 1,..., λ p := Ad 1 ck x p n=k = Adei k=1 ( p n=1 λ n x n 1 n i=1 (1 c, ix RAdV ( 1 c 1,..., 1 cp (λ 1,..., λ p = R AdV ( 1 c 1,..., 1 cp (λ 1,..., λ n RAdV ( 1 c,..., 1 1 cp (λ 1,..., λ p 1 c 1,..., 1 c p versal addition AdV 0 (a 1,...,a p (λ 1,..., λ p := Ad ( p k=1 λn c k 1 i n, i k (c k c i versal addition p n=k (x a k ( λ 1 λ 1 u = 0 x a 1 x a 2 a 1 = a 2 ( λ 1 + λ 2 u = 0 x a 1 λ 1 + λ 2 = µ 1 µ 2 + x a 1 x a 2 x a 1 (x a 1 (x a 2 a 1 = a 2 ( µ 1 µ 2 x a 1 (x a 1 2 u = 0 λn 1 i n,i k (a k a i x = a 1 versal addition 3.3 versal operator versal addition versal Jordan-Pochhammer P := RAd ( µ RAdV ( 1 c,..., 1 1 cp (λ 1,..., λ p ( p = RAd ( µ λ k x k 1 R + k k=1 ν=1 (1 c νx = µ+p 1( p 0 (x + q(x p µ = p k (x p k k=0

27 24 3 p 0 (x = p (1 c j x, q(x = p λ k x k 1 k=1 p j=k+1 (1 c j x, ( ( µ + p 1 p k (x = p (k µ + p 1 k 0 (x + q (k 1 (x k 1 c j c j, c i 0 Jordan-Pochhammer versal Jordan-Pochhammer p = 2 Gauss versal Gauss P c1,c 2 ;λ 1,λ 2,µ = RAd ( µ RAdV ( 1 c, 1 1 c 2 (λ 1, λ 2 ((1 c 1 x λ 1 = RAd ( µ RAd c 1 + λ 2 c 1 (c 1 c 2 (1 c 2 x λ 2 c 2 (c 2 c 1 ( = RAd ( µ RAdei λ 1 1 c 1 x λ 2 x (1 c 1 x(1 c 2 x ( = RAd ( µ R + λ 1 1 c 1 x + λ 2 x (1 c 1 x(1 c 2 x = Ad( µ ( (1 c 1 x(1 c 2 x + (λ 1 (1 c 2 x + λ 2 x = ( (1 c 1 x + c 1 (µ 1 ( (1 c 2 x + c 2 µ + λ 1 + (λ 2 λ 1 c 2 (x + 1 µ = (1 c 1 x(1 c 2 x 2 + ( (c 1 + c 2 (µ 1 + λ 1 + (2c 1 c 2 (1 µ + λ 2 λ 1 c 2 x + (µ 1(c 1 c 2 µ + λ 1 c 2 λ 2 = (1 c 1 x(1 c 2 x 2 + ( λ 1 + λ 2 x + µ ( λ2 c 1 c 2 ( µ + 1, λ 1 = λ 1 + (c 1 + c 2 (µ 1, λ 2 = λ 2 λ 1 c 2 + 2c 1 c 2 (1 µ, µ = 1 µ. c 1 c 2, c 1 c 2 0 Gauss Riemann λ 1 c 1 + x = 1 c 1 1 c µ ; x λ 2 c 1 (c 1 c 2 + µ λ 2 c 2 (c 2 c 1 + µ λ 1 c 1 + λ 2 c 1 c 2 µ K c1,c 2 ;λ 1,λ 2 (x I µ c K c1,c 2 ;λ 1,λ 2 (x = 1 Γ(µ x K c1,c 2 ;λ 1,λ 2 (x = ( 1 c 1 x λ 1 c c 1 + K c1,c 2 ;λ 1,λ 2 (t(x t µ 1 dt λ 2 c 1 (c 2 c 1 ( 1 c 2 x λ 2 c 2 (c 2 c 1

28 3.3 versal operator 25 c versal Gauss versal addition P c1,0;λ 1,λ 2,µ = (1 c 1 x 2 + ( c 1 (µ 1 + λ 1 + λ 2 x λ 2 (µ 1. Kummer *1 K c1,0;λ 1,λ 2 (x = ( 1 c 1 x λ 1 c 1 + λ 2 c 2 1 exp( λ2 x c 1 P 0,0;0, 1,µ = 2 x + (µ 1 Hermite Weber Weber Ad(e 1 4 x2 P 0,0;0,1,µ = ( 1 2 x2 + x( 1 2x (µ 1 ( 1 = µ x2 4 ( x K 0,0;0,±1 = exp 0 ±t dt = exp D µ (x := ( 1 µ e x2 4 I µ (e x2 e x = Γ(µ = e x 2 4 Γ(µ e (s+x2 0 x µ e x2 4 2F 0 (µ = k=0 x µ e x2 4 x 2 s µ 1 ds = e x 2 4 Γ(µ 2, µ , ; 2 x 2 ( µ 2 k( µ k k! ( 2x 2 k (± x2 2 e t2 2 (t x µ 1 dt 0 e xs s2 2 s µ 1 ds x + n = 2 versal Jordan-Pochhammer c 1 = = c p = 0 u(x = C e λ j t j 1 dt (x t µ 1 dt = C ( p exp λ j j tj (x t µ 1 dt *1 RAd ( µ RAdV 0 0,a (λ 1, λ 2 a = 0 Kummer

29 26 3 C x λ p 0 e λ p+ 2π 1k p (k = 0, 1,..., p 1 p λ p = 1, p = 3 λ p = 1, p = 4 x C 0 0 x C 2 V c = V c = p c j C 1 c j x, p C h j (x, h j (x = x j 1 j i=1 (1 c ix. c j 0 (j = 1,..., p, c i c j (i j V c = V c Ad ( p (x 1 c j µ j Ad ( p (1 c jx µ j + p AdV ( 1 c 1,..., 1 cp (µ 1,..., µ p + µ j c j 1 c j x, x x p µ jh j (x, x x Ad V c AdV V c c j Ad AdV c j c j, c i c j = 0 c j = c i V c V c c j ((h 1,..., h p. (c 1,..., c p h 1 (x,..., h p (x Ṽ c = p C h j c 1,..., c p p c j 0 (j = 1,..., p c i c j (i j V c = Ṽc h i = p g ij(ch j (1 i p c = (c 1,..., c p C p g ij (c det ( g ij (c 1 i p 1 j p

30 t α 1 (1 t β 1 dt = Γ(αΓ(β Γ(α + β. Re α > 0, Re β > 0 (1 t γ ( γ( γ 1 ( γ ν + 1 = ( t ν ν! ν=0 Γ(γ + ν = t ν (γ ν = t ν Γ(γν! ν! ν=0 Euler I µ c I µ c u(x = 1 Γ(γ = x c (x cµ Γ(µ I µ c I µ c = I µ+µ c, Ic n u(x = dn dx n u(x, I µ c (x c λ = 1 Γ(µ = = x c (x cµ Γ(µ (x t µ 1 u(t dt 1 0 ν=0 (1 s µ 1 u ( (x cs + c ds (t = (x cs + c, (t c λ (x t µ 1 dt 1 (x cλ+µ Γ(µ (1 s µ 1( (x cs λ ds (1 s µ 1 s λ ds Γ(λ + 1 = Γ(λ + µ + 1 (x cλ+µ, c n (x c λ+n Γ(λ + n + 1 = Γ(λ + µ + n + 1 c n(x c λ+µ+n. n=0 n=0

31 28 4 λ λ / { 1, 2,,...} c = I µ n=0 c n x λ n = e πiµ n=0 Γ(λ µ + n c n x λ µ n. Γ(λ + n O c n=0 c n(x c n (lim n c n 1 n < λ / { 1, 2,...} O c (λ, m = (x c λ m j=0 O c log j (x c I µ c : O c (λ, m O c (λ + µ, m well-defined φ j O c (j = 0,..., m I µ c ( m (x c λ φ j log j (x c I c µ ( (x c λ φ m log m (x c O c (λ + µ, m 1. j=0 λ + µ / { 1, 2,...} I µ c Proof. c = 0 φ O 0 Γ(λ+n m = 0 lim n Γ(λ+µ+n (λ + µn = 1 well-defined *1 I µ 0 I µ ( 0 I c µ x λ φ(x λ m > 0 I µ ( 0 x λ φ(x log m x = dm ( dλ m Iµ c x λ φ(x well-define P W [x], φ O c (λ, m P φ = 0 k P = ϑ ϑ µ Q = i 0, j 0 i 0, j 0 c ij i ϑ j (c ij C c ij i (ϑ µ j S, T W [x] Q = ST S x = c λ + µ / Z T (I µ c (φ = 0. *1 N N λ λ + N I µ 0 = dn dx N I µ+n 0

32 ( RAd ( µ P ( I µ c (φ = 0 ( RAd ( µ P u = 0 W (x ( RAd ( µ P = {T W (x T I µ c (u = 0}. (4.1 Proof. Ad( µ ϑ = ϑ µ P φ = 0 Q ( I µ c (φ = µ+k P µ I µ c (φ = µ+k P φ = 0 ST I µ c φ = 0 φ O c (λ, m I µ c (φ O c (λ + µ, m N 0 T I µ c (φ O c (λ + µ N, m λ + µ N / Z Sv = 0 x = c S ( T I µ c (φ = 0 T I µ c (φ = 0 (4.1 W (x Gauss x 1 γ I γ β ( 0 x β 1 (1 x α 2 *7 x 1 γ I γ β ( 0 x β 1 (1 x α = x 1 γ I γ β 0 = ν=0 ν=0 Γ(β + ν Γ(γ + ν (α ν x β 1+ν ν! (α ν x ν = Γ(β ν! Γ(γ ν=0 (α ν (β ν x ν. (γ ν ν! P W [x] m λ / Z, λ + µ / Z λ, µ O(λ, m φ P u = 0 P j, S j W [x] N P js j W [x]p l Z φ j = S j φ, Q j = l µ P j µ W [x], (j = 1,..., N (4.2 N ( Q j I µ c (φ j = 0 Proof. l Z j = 1,..., N l P j = P j (, ϑ = i 1,i 2 c i1,i 2 i 1 ϑ i 2 (c i1,i 2 C N l P j S j φ = 0 N ( Iµ c Pj (, ϑs j φ = N P j (, ϑ µi c µ (S j φ = 0

33 (Gauss. u λ1,λ 2,µ := I µ ( 0 x λ 1 (1 x λ 2 Gauss Gauss I µ c = I µ 1 c u λ1,λ 2,µ 1 = u λ1,λ 2,µ, u λ1 +1,λ 2,µ = (x + 1 µu λ1,λ 2,µ, u λ1,λ 2 +1,µ = ( (1 x + µ 1 u λ1,λ 2,µ φ = x λ 1 (1 x λ 2 x 1 φ 1 x φ = 0 S 1 = T 2 = 1 T 1 = S 2 = x 0 = 1 µ x µ I µ 0 (φ 1 µ 1 µ I µ 0 (xφ = (x + 1 µi µ ( 0 x λ 1 (1 x λ 2 I µ ( 0 x λ 1 +1 (1 x λ Q W (x MP λ1,λ 2,µ + NQ = 1 M, N W (x Q = R = x(1 x + 1 λ 1 µ + (λ 1 + λ 2 + 2µ 2x P λ1,λ 2,µ = R (λ 1 + λ 2 + µ(µ 1 M, N Ru λ1,λ 2,µ 1 = (λ 1 + λ 2 + µ(µ 1u λ1,λ 2,µ k 1, k 2, k 3 u λ1 +k 1,λ 2 +k 2,µ+k 3 = T (k 1, k 2, k 3 u λ1,λ 2,µ 1 T (k 1, k 2, k 3 W (x; λ 1, λ 2, µ 3 u(k 1, k 2, k 3 := u λ1 +k 1,λ 2 +k 2,µ+k 3 (k 1, k 2, k 3, (l 1, l 2, l 3 Z 3 3 a 0 u(0, 0, 0 + a 1 u(k 1, k 2, k 3 + a 2 u(l 1, l 2, l 3 = 0 a j C(x, λ 1, λ 2, µ a 0, a 1, a 2 0 u(k 1, k 2, k 3 = (b 1 + c 1 u(0, 0, 0, u(l 1, l 2, l 3 = (b 2 + c 2 u(0, 0, 0 b 1, c 1, b 2, c 2 C(x, λ 1, λ 2, µ

34 31 5 Fuchs Riemann 5.1 Fuchs P = a n (x dn dx n + a n 1(x dn 1 dx n a 0(x ( ai (x O 0 ( x = c C P j (1 j n a n j(x a n (x x = c x = c b j (x := (x c n j a j(x a n (x O c (j = 0, 1,..., n a j(x a n (x x = c n j P x = 0 x n a n (x P = xn n + b n 1 (xx n 1 n b 0 (x x k k = ϑ(ϑ 1 (ϑ k + 1 x ϑ x n a n (x P = P 0(ϑ xr(x, ϑ R(x, ϑ = r n 1 (xϑ n r 1 (xϑ + r 0 (x ( rj (x O 0, P 0 (ϑ = x n n + b n 1 (0x n 1 n b 0 (0

35 32 5 Fuchs Riemann, P 0 (ϑ ϑ P 0 (ϑ P 0 (s := s(s 1 (s n b n 1 (0s(s 1 (s n b 0 ( P 0 (s = 0 λ 1,..., λ n P x = 0 r B r = {x C x < r} Ô0 = C[[x]] Ô0 O(B r. φ = a j x j, ψ = b j x j Ô0 ψ φ a j b j j φ ψ i φ ψ ψ O(B r φ O(B r ii φ O(B r Cr > 1 C > 0 A C > 0 φ A C 1 Cx = j=0 A C (Cx j Ô x = 0 P P = P 0 (ϑ xr(x, ϑ deg P 0 (s = n > ord R j r j (x O(B r r > 0 P 0 (k 0 (k = 0, 1, 2,... P : O(B r O(B r Proof. f = f j x j Ô0 P u = f u = u j x j j=0 j=0 P 0 (ϑ u j x j = j=0 = P 0 (ju j x j = j=0 f j x j + xr(x, ϑ u j x j j=0 j=0 = f j x j + x R(x, ju j x j j=0 j=0 x j u j

36 f O(B r u O(B r O(B r O(B r P 0 (k 0 ɛ > 0 n j P 0 (k ɛ (k + 1 ν j=0 ν= Cr > 1 C > 0 A, B > 0 j=0 b j (x A 1 Cx f B 1 Cx (j = 0,..., n 1 n 1 ( xr(x, ϑ = bj (0 b j (x x j j, b j (0 b j (x A 1 Cx A = ACx 1 Cx û = j=0 ûjx j ( n ɛ x j j û j=0 n 1 j=0 ACx 1 Cx xj j û + B 1 Cx (5.2 û u n 1 û j ACx 1 Cx xj j û + B 1 Cx xj. j=0 û 0 u 0 l j û j u j l û j x j j=0 l u j x j j=0 n 1 B 1 Cx + ACx 1 Cx xj j j=0 l û k x k f + xr(x, ϑ k=0 l u k x k k=0 φ φ, ψ ψ φ + ψ φ + ψ, φ ψ φψ, ϑ φ ϑφ û l+1 u l+1 û u (5.2 û N, L ɛ n x j j=0 û := L (1 Cx N C j n 1 (N j L (1 Cx j+n ACx C j (N j L 1 Cx xj û (5.2 (1 Cx j+n = n L ɛ (1 Cx N B 1 Cx x j ɛcj (N j L AC j (N j 1 L (1 Cx j+n 0,

37 34 5 Fuchs Riemann m O Br (λ, m = m j=0 O(B r x λ log j x P = P 0 (ϑ xr(x, ϑ P 0 (λ + k 0 (k = 0, 1, 2,... P : O Br (λ, m O Br (λ, m Proof. P λ = Ad(x λ P P u = x λ P λ x λ u, P λ = P 0 (ϑ + λ + xr(x, ϑ + λ λ = 0 ( φ(x O(B r P λ log m x φ(x ( P λ φ(x log m x mod O Br (0, m m (Cauchy-Kowalevskaja *1. n 1 Q = n + q j (x j ( qj (x O(B r j=0 Q: O(B r O(B r C n u ( Qu, u(0, u (0,..., u (n 1 (0 Proof. P = Qx n P = Qx n = n q j (x j x n = j=0 n q j (x(ϑ + 1 (ϑ + jx n j j=0 P 1, 2,..., n P : O(B r O(B r (f, v 0, v 1,..., v n 1 O(B r C n w O(B r Qx n w = f Q (v 0 + x 1! v xn 1 (n 1! v n 1 u = x n w + v 0 + x 1! v xn 1 (n 1! v n 1 ( Qu, u(0,..., u (n 1 (0 = ( f, v 0,..., v n 1 u O(Br u (j (0 = 0 (j = 0,..., n 1 u = x n w w O(B r Qu = 0 P w = Qx n w = Qu = 0 w = 0 u = 0 *1 Cauchy-Kowalevskaja [O7] [O1]

38 P C { } U U U x 0 P u = 0 u(x x 0 U γ Proof. γ [0, 1] U, t γ(t γ 0 t < t 0 t = t 0 0 < t 0 < 1 x t 0 < r x U r > 0, t 1 t t 0 γ(t γ(t 0 < r 3 0 < t 1 < t 0 t 1 u t = t 1 u 1 P v = 0 v (j( γ(t 1 = u (j ( 1 γ(t1 (j = 0,..., n 1 γ(t 1 2r 3 γ(t 1 u 1 u(t γ γ(t 0 n P u = 0 x = 0 λ 1,..., λ n i 1 i < j n i, j λ i λ j / Z x λ j φ j (φ j O 0, φ j (0 0, j = 1, 2,..., n ii = 0 (1 j k, λ j λ 1 Z >0 (k < j m, / Z 0 (m < j n *2 u ν (x (ν = 0, 1,..., k 1 u ν x λ 1 log ν x O Br (λ 1 + 1, m 1. { λ 1,..., λ q } λ i m i n = m m N ū i,ν x λ i log ν x O Br ( λ i + 1, n 1 (0 ν < m i, i = 1,..., q (5.3 ũ i,ν (0 ν < m i, i = 1,..., q x = 0 Proof. P = P 0 (ϑ xr(x, ϑ i P = x λj 1 P x λj+1 P = P 0 (ϑ + λ j + 1 xr(x, ϑ + λ j + 1 x = 0 λ ν λ j 1 (ν = 1,..., n P 0 (ϑ ϑ + 1 = x xr(x, ϑ+λ j +1 = R(x, ϑ+λ j x O(B r Q P = Qx P P w = Q1 w O(B r P ( x λ j (1 xw = x λj+1 Qx x λ j 1 ( x λ j (1 xw = x λj+1 Q(1 xw = x λj+1 (Q1 P w = 0 *2 Z >0 := {1, 2, 3,...}, Z 0 := {0, 1, 2, 3,...}

39 36 5 Fuchs Riemann ii x λ 1 P x λ 1 λ 1 = 0 P 0 (ϑ = ϑ k (ϑ λ k+1 (ϑ λ n 0 j < k j P 0 (ϑ log j x = 0 P log j x = P 0 (ϑ log j x xr(x, ϑ log j x O Br (1, j P u = 0 u O Br (0, m P 0 (ϑ ( m log j x + u i,ν x i log ν x = xr(x, ϑ ( m log j x + u i,ν x i log ν x i=1 ν=0 i=1 ν=0 u i,j C, i Q = q(ϑ(ϑ λ l q(s q(λ 0 Q: n+l n Cx λ log j x Cx λ log j x, j=0 j=0 f Qf N 1 i N u i,ν C N m P u N O Br (N + 1, m, u N := log j x + u i,ν x i log ν x i=1 ν=0 N = max{0, λ k+1,..., λ m } P u N O Br (N + 1, m u = u N v P u = 0 = P v v 5.2 Fuchs P u = 0 x = x 1 x P W (x C := C { } P P u = 0 Fuchs C P {c 0, c 1,..., c p } C C := C\{c 0, c 1,..., c p } P F : C U( F(U O(U, F(U := {u(x O(U P u(x = 0} {c 0, c 1,..., c p } { {c j + re 1θ 0 < r < ɛ, R < θ < R + 2π} (c j, U j,ɛ,r = {re 1θ r > ɛ 1, R < θ < R + 2π} (c j = F 1. F(U O(U dim F(U = ord P. 2. V U F(V = F(U V = {f V O(V f F(U} 3. c j ɛ > 0 φ F(U j,ɛ,r C m φ(x < { C x c j m (c j, x U j,ɛ,r, C x m (c j =, x U j,ɛ,r

40 5.2 Fuchs 37 R R F W (x B + r := {x B r Im x > 0} ψ O(B + r m Proof. 0 < x < r 2 ψ(x = ψ(x C x m (x B + r ψ (x 2 m +1 Cx (m+1 (0 < x < r 2 1 2π 1 t x = x 2 ψ(tdt t x ψ 1 (x = 2π t x = x 2 ψ(tdt (t x 2 max ψ(t 2 t x = x 2 x 2 m +1 Cx (m Fuchs {W (xp W (x : Fuchs } { 1 3 F } P = n + n 1 i=0 a i(x i {U {u O(U P u = 0}} F(U = n ν=1 Cφ ν(x Φ i = φ (0 1 φ (i 1 φ (i+1 (x φ(0 n... 1 (x φ (i 1 n (x 1 (x φ (i+1 n (x... φ (n 1 (x φ(n n (x a i (x = ( 1 n i det Φ i det Φ n (5.4. φ ν (x (ν = 1,..., n F(U u(x F(U u φ 1 φ n u (1 φ (1 1 φ (1 n det.... = 0 u (n φ (n 1 φ (n n

41 38 5 Fuchs Riemann 1 u(x Fuchs c j γ j Φ i γ j γ j Φ i γ j φ 1,..., γ j φ n F(U φ 1,..., φ n F(U M j GL(n, C γ j Φ i = Φ i M j M j Φ i (i = 1,..., n γ j det Φ i = det M j det Φ i det M j = e 2πiλ λ C γ j ( (x cj λ det Φ i = (x cj λ det Φ i. (x c j λ det Φ i x = c j det Φ i ψ j O cj k i Z 0 det Φ i = (x c j λ+k i ψ i det Φ i det Φ n (x c j λ a i (x x = c j x = c j 3 cf c 0,..., c p C n Fuchs P u = 0 x 0 C V 0 V 0 φ 1,..., φ n x 0 c j γ j φ i γ j φ i V 0 V 0 φ 1,..., φ n V 0 V 0 v γ j v (γ j φ 1,..., γ j φ n = (φ 1,..., φ n M j M j GL(n, C M j M = (M 0,..., M n P u = 0 *3 Fuchs Wronskian det Φ n (x {k 1,..., k n } k j 0 k 1 < k 2 < < k n k n = n * n Fuchs P u = 0 1. P *3 M GL(n, C *4 Airy

42 5.2 Fuchs M = (M 0,..., M p M j V V (j = 0,..., p C n V {0} C n 3. F F {0} F (U F(U c j = 0 i n M, exp L := ν=0 Lν ν! = M L ii F u = (u 1,..., u n c j = 0 M i L v = ux L, x L := exp( L log x v c j c j iii c j = 0 F (5.3 iv x n i a i (x a n (x x = 0 x = 0 Fuchs W (x * P, Q W (x Fuchs W (x/w (xp W (x/w (xq W (x P u = 0 Qv = P = n + n 1 i=0 a i(x i W (x P y = y y Y =., A P (x = y (n a 0 (x a 1 (x a 2 (x a n 1 (x d dx Y = A P (xy P W (x W (x/w (xp W (x P u = 0 u W (x/w (xp W (x/w (xp C(x u, u,..., n 1 u W (x/w (xp C(x v W (x/w (xp v = t A p (xv W (x/w (xp P *6 *5 (x 2F (α, β, γ; x x = 2 *6 M P = W (x/w (xp M P MP f M P f(m = f( m (m M P W (x MP {u, u,..., n 1 u}

43 40 5 Fuchs Riemann. W (x/w (xp W (x/w (xq C(x dim C(x W (x/w (xp = dim C(x W (x/w (xq = n. P y = 0 C(x Y = A(xY, (5.5 Qz = 0 Z = B(xZ (5.6 W (x/w (xp W (x/w (xq W (x C(x G(x GL ( n, C(x Z = G(xY Y = A(xY Z = G(xY Z = B(xZ U C (5.5 y 1,..., y n ( O(U n (5.6 z1,..., z n ( O(U n Y = (y 1 y n GL ( n, O(U Z = (z 1 z n GL ( n, O(U ZY 1 GL ( n, C(x c j γ j γ j Y = YM j, γ j Z = ZN j M j, N j GL(n, C ZY 1 GL(n, C(x γ j ZY 1 = ZY 1 (j = 0,..., p ZN j M 1 j Y 1 = ZY 1 (j = 0,..., p M j = N j (j = 0,..., p 5.3 x = 0 3 P ε u = 0, P ε = ϑ(ϑ 1 2 (ϑ 1 εx x2 (ε = 0, 1 0, 1 2, x 1 2 (1 + a1 x + a 2 x 2 +, x(1 + b 1 x + b 2 x + A P (x W (x F P u = 0 F Hom W (x (M P, F Hom W (x (M P, F Hom C(x (M P, F = M P C(x F Hom W (x (M P, F = {u f M P F u f = u f} A P (x

44 5.3 41, 0 { 1 + c 2 x 2 + c 3 x 3 + (ε = 0, 1 + 2x log x + (c 2 + c 2 log xx 2 + (c 3 + c 2 log xx 2 + (ε = 1 ɛ = πi 1, * O 0 P = a n (x n + a n 1 (x n a 0 (x (5.7 x = 0 a n (x x = 0 n k 0 k n a n (0 = = a (n 1 n (0 = 0, a (n n (0 0. ( O 0 R P = x k R. 2. P x ν = o(x k 1 ν = 0, 1,..., k 1 3. P u = 0 ν = 0, 1,..., k 1 u(x = x ν + o(x k 1 4. P P = j 0 x j p j (ϑ x ϑ p j (ϑ p j (ν = 0 0 ν < k j, j = 0,..., k ν = 0, 1,..., k 1 p 0 (ν = 0 0, 1,..., k 1 3 log Proof P x ν = j 0 xj p j (ϑx ν = j 0 xj+ν p j (ν. j + ν < k p j (ν = 0 P x ν = o(x k P 1 = o(x k 1, P x = o(x k 1,..., P x k 1 = o(x k 1 *7 ε C \ {0}

45 42 5 Fuchs Riemann j = 0,..., k 1 b j (x O 0 a j (x = x k b j (x x = 0 j = k, k + 1,..., n b j (x O o a j (x = x k b j (x O 0 n P = n i=0 a i(x i x = 0 P x = 0 [λ] (k x n k Ad(x λ ( a n (x 1 P W [x] P x = 0 [λ] (k q l (x x n a n (x 1 P = l 0 x l q l (ϑ 0 i<k l (ϑ λ i n x = 0 [0] (n x = 0 [0] (n (. O 0 n P n = m m q m i λ 1,..., λ q P x = 0 {[λ 1 ] (m1,..., [λ q ] (mq } x n a n (x 1 P = l 0 x l q l (ϑ q ν=1 0 i<m ν l (ϑ λ ν i *8 {m 1,..., m q } P x = 0 x = c x = c P x = 0 [λ] (m m = 1 m λ, λ + 1,..., λ + m 1 x λ+m 1 log ( Riemann. Fuchs P W [x] c 0 =, c 1,..., c p x = c j P {[λ j,1 ] (mj,1,..., [λ j,nj ] (mj,nj } x = c 0 c 1 c p [λ 0,1 ] (m0,1 [λ 1,1 ] (m1,1 [λ p,1 ] (mp,1 {λ m } :=.... [λ 0,n0 ] (m0,n0 [λ 1,n1 ] (m1,n1 [λ p,np ] (mp,n p Riemann (generalized Riemann scheme (5.9 *8 P (5.8 P = l 0 xl q l (ϑ q ν=1 0 i<m ν l (ϑ λ ν i

46 (. P W [x] Riemann n (p + 1 m := (m 01,..., m 0,n0 ; m 1,1,..., m 1,n1 ;... P P λ j,ν λ j,ν / Z (ν ν (semi-simple (Fuchs. n p j m j,ν 1 j=0 ν=1 i=0 Proof. n 1 P (λ j,ν + i = (p 1n(n 1 2 (5.10 P = n p a j x c j n 1 + a j C x = n 1 x = 0 c j c j c j = 0 x n P x n n a j x n 1 n 1 = ϑ(ϑ 1 (ϑ n + 1 a j ϑ(ϑ 1 (ϑ n + 2 ( (n 1n = ϑ n + a j ϑ n c j n j ν=1 m j,ν 1 i=0 (λ j,ν + i = a j + (n 1n 2 (1 j p c 0 = y = x 1 ϑ ϑ y x n n p a j x n 1 n 1 = ϑ(ϑ 1 (ϑ n + 1 ( (n 1n = ( 1 (ϑ n n y c 0 = n 0 ν=1 m 0,ν 1 i=0 (λ 0,ν + i = p a j Fuchs p a j ϑ(ϑ 1 (ϑ n + 2 p a j ϑ n 1 y + (n 1n 2

47 44 5 Fuchs Riemann Gauss (1, 1; 1, 1; 1, 1 11; 11; 11 11, 11, ; 1 2 ; ( n F n 1 Jordan-Pochhammer (n 11; 1 n ; 1 n (n+1 {}}{ (n 11; (n 11; n n n m = ( m j,ν j=0,1,... m j,ν m ν=1,2,... m j,ν = n (j = 0, 1,..., ν=1 m j,1 = n (j > p m n (p + 1 n j ν n j m j,ν = 0 m = (m j,ν j=0,...,p ν=1,...,n j n (p + 1 m j,ν m j,ν+1 j = 0, 1,... ν = 1, 2,... ( m j,ν j=0,1,... ν=1,2,... gcd m m gcd m = 1 n (p + 1 P (n p+1 P p+1 = n=0 P (n p+1, P(n = p=0 m P (n m ord m = n P (n p+1, P = p=0 P p+1

48 m, m P p+1 idx (m, m := p j=0 ν=1 m j,ν m j,ν (p 1ord m ord m m = m idx m = idx (m, m m rigidity Pidx m = 1 idx m 2 Fuchs P p+1 m = ( m j,ν 0 j p ( λ j,ν 0 j p 1 ν n j 1 ν n j {λ m } := n p j j=0 ν=1 m j,ν λ j,ν ord m + idx m 2 (5.11 {λ m } Fuchs (5.10 {λ m } = 0 (5.12 n p j j=0 ν=1 m j,ν 1 i=0 i = 1 2 = 1 2 n p j m j,ν (m j,ν 1 = 1 2 j=0 ν=1 ( idx m + (p 1n 2 1 (p + 1n 2 = 1 (p 1n(n 1 idx m n n p j m 2 j,ν 1 (p + 1n 2 j=0 ν=1 5.5 m = (m 1,..., m N P (n 1 λ = (λ 1,..., λ N C N n n L(m; λ m m 1 m n A i,j M(m i, m j, C λ i I mi (i = j, A i,j = I mi,m j = ( ( I mj δ µν 1 µ m i = (i = j 1, 1 ν m j 0 0 (i j, j 1

49 46 5 Fuchs Riemann L(m; λ = ( A i,j 1 i N 1 j N L(2, 1, 1; λ 1, λ 2, λ 3 = λ λ λ λ 3 m *9 A M(n, C A L(m; λ j rank (A λ ν = n (m m j j = 0, 1,..., N (5.13 ν=1 L(m; λ Jordan A L(m; λ Jordan A M(n, C Z(A := {X M(n, C; AX = XA} dim Z ( L(m; λ = m m 2 N m m = (m 1,..., m r m ν = #{j m j ν} ( : (7, 6, 3, 1 (4, 3, 3, 2, 2, 2, 1 m µ 1 µ 1 J(k, µ := M(k, C 1 µ Jordan L(m; λ Jordan m µ = (µ,..., µ C N m 1 L(m; µ J(m j, µ ( *9 L(1, 2, 1; λ 1, λ 2, λ 3 = L(2, 1, 1; λ 2, λ 1, λ 3

50 J(k, µ l k dim{u C k J(k, µ l 1 u 0, J(k, µ l u = 0} = 1 J = m 1 J(m j, µ dim{u C n J l 1 u 0, J l u = 0} = #{ν m ν l} = m l. J ( n P u = 0 x = 0 {[λ 1 ] (m1,... [λ q ] (mq } n = m m q i k q λ 1 = λ 2 = = λ k, m 1 m 2 m k, λ j λ 1 / {0, 1, 2,...} (j = k + 1,..., q m = {m 1,..., m r } m = {m 1,..., m k } r = m 1 i = 0, 1,..., r 1 j = 0,..., m i+1 1 j u i,j (x = x λ1+i log ν x φ i,j,ν (x u i,j φ i,j,ν (x O 0 φ i,j,ν (0 = δ j,ν ii λ i λ j / Z \ {0} (i j * 10 x = 0 L ( m; e(λ 1,..., e(λ q ν=0 λ i λ j / Z (i j Proof. ii i 1 λ 1 = 0 P P = x l r l (ϑ l=0 k ν=1 0 i m ν l (ϑ i r l (s r 0 (s = 0 P = l=0 xl p l (ϑ 0 i m 1 1 i p l (x = 0 *10 [O6, Proposition 11.9] λ i λ j / Z (λ i λ j (λ i + m i λ j m j (, 0]

51 48 5 Fuchs Riemann m i+l+1 (ν m 1 m ν = 0 p 0 (s = 0 i m i+1 i, j x l p l (ϑx i log j x = x i+l j m i+l+1 ν=0 c i,j,l,ν log ν x (c i,j,l,ν C p 0 (ϑx i log j x = 0 (j < m i+1 p 0 (ϑ : Cx i log ν x ν j ν j m i+1 Cx i log ν x (5.14 p 0 (ϑφ i,j = 1 l<m 1 i x l p l (ϑx i log j x, φ i,j φ i,j x i log j x φ i,j V := Cx i log j x 0 i<m 1 0 j<m 1 i<k<m 1 0 ν j Cx k log ν x Q u V T u = m 1 1 ν=0 Q ν u m 1 1 P T u p 0 (ϑt u + x l p l (ϑt u mod O 0 (m 1, j l=1 p 0 (ϑ(1 QT u mod O 0 (m 1, j p 0 (ϑ(1 Q(1 + Q + + Q m 1 1 u mod O 0 (m 1, j = p 0 (ϑu j < m i+1 P T xi log j x 0 mod O 0 (m 1, j v i,j = T x i log j x P v i,j O 0 (m 1, j w i,j O 0 (m 1, j P v i,j = P w i,j u i,j = v i,j w i,j 5.6 n n (. m = (m 0,..., m p n (p + 1 m j = (m j,1,..., m j,nj j = 0,..., p m j,ν m j,1 + + m j,nj = n c 1,..., c p c 0 = m Fuchs P Fuchs λ j,ν Riemann (5.9 P m

52 m, m P (n p+1 m + m c 1,..., c p C c 0 = x = c j (j = 0,..., p Fuchs a i (x ( p P = (x c j n n + a n 1 (x n a 1 (x + a 0 (x (5.15 deg a i (x (p 1n + i, ( ν a i (c j = 0 (0 ν i 1, 1 j p (5.16 ( P Riemann (5.9 P P c 0,..., c p Riemann c 0 = c j (j = 1,..., p [λ 0,1 + m 0,1 (p 1 ord m] (m 0,1 [λ j,1 + m j,1 ] (m j,1.. [λ 0,n0 + m 0,n0 (p 1ordm] (m 0,n0 [λ j,nj + m j,n ] (m j,nj (5.17 P P n m P P m + m Riemann c 0 c 1 c p [λ 0,1 ] (m0,1 +m 0,1 [λ 1,1 ] (m1,1 +m 1,1 [λ p,1 ] (mp,1 +m p,1.... [λ 0,n0 ] (m0,n0 +m 0,n [λ 1,n1 ] (m1,n1 +m 0 1,n [λ p,np ] (mp,n 1 p +m p,np (5.18 (5.9 Fuchs (5.17 Fuchs (5.18 Fuchs { } λ m + { } [λ j,ν + m j,ν δ j,0 (p 1 ord m] (m j,ν 0 j p = { } λ m+m. 1 ν n j Proof. Q = Q = x l r l (ϑ l=0 x l r l(ϑ l=0 q ν=1 0 k<m ν l q ν=1 (ϑ λ ν k, 0 k<m ν l (ϑ λ ν m ν k, r l (s, r l (s (ϑ λ ν m ν k x i (ϑ λ ν k = x i (ϑ λ ν m ν k 0 k<m ν j 0 k<m ν i 0 k<m ν +m ν i j

53 50 5 Fuchs Riemann Q Q = x l( l q r l i(ϑ + ir i (ϑ l=0 i=0 ν=1 0 k<m ν +m ν l (ϑ λ ν k x = c 1,..., c p P P Riemann y = x c j x = c 0 = Q = x (p 1 ord m P, Q = x (p 1 ord m P Q Q = x 2(p 1 ord m Ad (x (p 1 ord m (P P y = 1 x (5.17 Fuchs n p j j=0 ν=1 m ( j,ν λj,ν + m j,ν δ j,0 (p 1 ord m = ord m idx m 2 (5.19 n p j m j,ν( mj,ν δ j,0 (p 1 ord m = idx (m, m j=0 ν=1 (5.19 n p j m j,νλ j,ν = ord m j=0 ν=1 idx m 2 idx (m, m idx (m + m = idx m + idx m + 2idx (m, m (5.9 Fuchs n p j (m j,ν + m j,νλ j,ν = ord(m + m idx (m + m 2 j=0 ν=1 (5.18 Fuchs

54 51 6 Fuchs 2 RAd ( (x c λ RAd ( µ Fuchs Riemann 6.1 Riemann i P u = 0 Fuchs c P ( cw [x] c = 0 ii φ(x C(x λ, µ C P W [x] P C[x]RAdei ( φ(x RAdei ( φ(x P, P C[ ]RAd ( µ RAd ( µ P P ord P > 1 RAd ( µ RAd ( µ P = cp c C Proof. i P = ( cq Qu(x = e cx u(x P u(x = 0 P x = N > 0 C > 0 x 1 0 arg x 2π u(x < C x N c = 0 ii Adei ( φ(x Adei ( φ(x = id, Adei ( φ(x f(xp = f(xadei ( φ(x P L 1 R LP = cp (c C L 1 R LP = P R LP = L P f(x C(x LP = f(xl P LP W [x] f(x C[x]

55 52 6 Fuchs P = f( P P P C[x] f(x C f( 1 ord P > 1 Ad ( (x c τ RAd ( µ Riemann P u = 0 Riemann (5.9 Fuchs P (5.15 i (addition Ad ( (x c j τ P Riemann c 0 = c 1 c j c p [λ 0,1 τ] (m0,1 [λ 1,1 ] (m1,1 [λ j,1 + τ] (mj,1 [λ p,1 ] (mp, [λ 0,n0 τ] (m0,n0 [λ 1,n1 ] (m1,n1 [λ j,nj + τ] (mj,nj [λ p,np ] (mp,n p ii (middle convolution m j,1 = 0 λ j,1 = 0 (j = 1,..., p { µ = λ 0,1 1, d = d(m := p j=0 m j,1 (p 1n m j,1 d (j = 0,..., p, (6.1 { ν 1 m0,ν m 0,1 d + 2 m 1,1 m p,1 0 (6.2 λ 0,ν / {0, 1,..., m 0,1 m 0,ν d + 2}, { j 1, ν 2 mj,1 0 m j,ν m j,1 d + 2 (6.3 λ 0,1 + λ j,ν / {0, 1,..., m j,1 m j,ν d + 2}. p S = d Ad ( µ (x c j m j,1 P W [x] Riemann c 0 = c 1 c p [1 µ] (m0,1 d [0] (m1,1 d [0] (mp,1 d [λ 0,2 µ] (m0,2 [λ 1,2 + µ] (m1,2 [λ p,2 + µ] (mp,2.... [λ 0,n0 µ] (m0,n0 [λ 1,n1 + µ] (m1,n1 [λ p,np + µ] (mp,n p. (6.4 (6.5 S = RAd ( µ RP x = c j (j = 0,..., p λ j,1 + m j,1 P (6.4

56 6.1 Riemann 53 m 0,1 = d λ 0,1 / { d, d 1,..., 1 m 0,1 }. (6.5 iii ord P > 1 P ii (6.1 (6.2 (6.3 *1 Proof. i ii, iii ( p P = (x c j m j,1 P P W [x] (6.4 RP = P Q = (p 1n p m j,1 P (6.1 (p 1n p m j,1 0 1 j p j x = c j j = 1 c j = 0 P = n j=0 a j(x j deg a j (x (p 1n + j p m j,1 x m 1,1 P = N x N l r l (ϑ l=0 1 ν n 0 0 i<m 0,ν l (ϑ + λ 0,ν + i p N = (p 1n m j,1 = m 0,1 + m 1,1 d, j=2 r l (s r 0 0 N m 1,1 + 1 l N 1 ν n 0 0 i<m 0,ν l (ϑ + λ 0,ν + i / xw [x] (ϑ + λ 0,ν + i = ϑ (6.2 m 1,1 = 0 l m 0,1 d + 1 m 0,ν m 0,1 d + 1 λ 0,ν / {0, 1,..., m 0,1 m 0,ν d + 2} (1 ν n 0 P W [x] x = c 1 = 0 [0] (m1,1 N m 1,1 + 1 l N s l r l (ϑ = x l N+m 1,1 l N+m 1,1 s l (ϑ *1 Riemann (5.9 P ord P > 1 λ 0,ν0 + + λ p,νp Z m 0,ν0 + + m p,νp (p 1 ord m cf. [O6, Lemma 7.3] (6.5

57 54 6 Fuchs r l s l N m 1,1 P = x N m1,1 l s l (ϑ + l=0 N l=n m 1, ν n 0 0 i<m 0,ν l l N+m 1,1 s l (ϑ (ϑ + λ 0,ν + i 1 ν n 0 0 i<m 0,ν l (ϑ + λ 0,ν + i (6.6 Q = Ad ( µ Q = N l s l (ϑ l=0 N l s l (ϑ µ l=0 1 i m 0,1 l N m 1,1 = m 0,1 d m 0,1 Ad ( µ Q = m 0,1 1 + l=0 N 1 i N m 1,1 l (ϑ + i 1 i N m 1,1 l (ϑ + i x m 0,1 l s l (ϑ µ l=m 0,1 l m0,1 s l (ϑ µ 2 ν n 0 0 i<m 0,ν l 1 i m 0,1 d l 1 i m 0,1 d l 1 ν n 0 0 i<m 0,ν l (ϑ µ + i (ϑ µ + λ 0,ν + i (ϑ µ + i (ϑ µ + i (ϑ + λ 0,ν + i, 2 ν n 0 0 i<m 0,ν l 2 ν n 0 0 i<m 0,ν l x = { [1 µ](m0,1 d, [λ 0,2 µ] (m0,2,..., [λ 0,n0 µ] (m0,n0 } (ϑ µ + λ 0,ν + i (ϑ µ + λ 0,ν + i l = 0 λ 0,ν + i λ 0,1 + m 0,1 µ + λ 0,ν + i m 0,1 + 1 λ 0,1 + m 0,1 P x = 1 i m 0,1 d µ + i m 0,1 + 1 λ 0,1 / { d, d 1,..., 1 m 0,1 } x m 0,1 s 0 (ϑ µ + i (ϑ µ + λ 0,ν + i / W [x] 1 i m 0,1 d 2 ν n 0 0 i<m 0,ν m 0,1 1 Ad ( µ Q / W [x] m 0,1 m Ad ( µ Q W [x] m ii x = c 1 = 0 P P = m 1,1 l=0 m 1,1 l q l (ϑ 2 ν n 1 0 i<m 1,ν l (ϑ λ 1,ν i+ N l=m 1,1 +1 x l m 1,1 q l (ϑ 2 ν n 1 0 i<m 1,ν l (ϑ λ 1,ν i

58 6.1 Riemann 55 Q = Ad ( µ Q = m 1,1 l=0 + N l q l (ϑ N l=m 1,1 +1 N N l q l (ϑ µ l=0 2 ν n 1 0 i<m 1,ν l (ϑ λ 1,ν i l m 1,1 N l q l (ϑ (ϑ + i i=1 2 ν n 1 0 i<m 1,ν l 1 i l m 1,1 (ϑ µ + i (ϑ λ 1,ν i, 2 ν n 1 0 i<m 1,ν l N m 0,1 = m 1,1 d (ϑ µ λ 1,ν i / W [x] 2 ν n 1 0 i<m 1,ν l N l < m 0,1 x = c 1 m 0,1 Ad ( µ Q { [0](m1,1 d, [λ 1,2 + µ] (m1,2,..., [λ 1,n1 + µ] (m1,n1 } (ϑ µ λ 1,ν i 0 i m 1,ν m 1,1 + d 2 ϑ µ λ 1,ν i = ϑ + 1 (6.3 ii iii (6.1 P, N ii x m 1,1 P = N x N l r l (ϑ l=0 1 ν n 0 0 i<m 0,ν l (ϑ + λ 0,ν + i m 0,1 > N x m 1,1 P ϑ + λ 0,1 P m 0,1 N m 1,1 d = m 1,1 + N m 0,1 m 1,1 = N m 0,1 0. m 1,1 > N x m 1,1 P ϑ m 1,1 N m 0,1 d 0 m j,1 d 0 (6.2 P p l (x (l = 0, 1,..., N x m 1,1 P = N x N l p l (ϑ l=0 x = 0 [0] (m1,1 p l (ϑ (N m 1,1 + 1 l N 0 i<l N+m 1,1 (ϑ i = x l N+m 1,1 l N+m 1,1 x = [λ 0,ν ] (m0,ν p l (ϑ (0 l N m 1,1 0 i<m 0,ν l (ϑ + λ 0,ν + i P = N m 1,1 l=0 x N m 1,1 l r l (ϑ 0 i<m 0,ν l (ϑ + λ 0,ν + i + N l=n m 1,1 +1 l N+m 1,1 r l (ϑ (6.7

59 56 6 Fuchs r l (s m 0,ν m 0,1 d + 2 λ 0,ν {0, 1,..., m 0,1 m 0,ν d + 2} 0 l N m 1,1 = m 0,1 d m 0,ν > l λ 0,ν + (m 0,ν l 1 m 0,1 m 0,ν d m 0,ν l 1 (6.7 1 = m 0,1 d l + 1 > 0. x m 0,1 d l (ϑ + 1(ϑ + 2 (ϑ + λ 0,ν + m 0,ν l 1 m 0,1 d l = L λ 0,ν + m 0,ν l 1 = M M > L 0 x L (ϑ + 1(ϑ + 2 (ϑ + M = (ϑ + 1 L(ϑ + 2 L (ϑ + M Lx L W [x] (6.7 2 P (6.3 P = Ad ( (x c j λ j,ν P (6.2 P d d d = m 0,1 + + m j 1,1 + m j,ν + m j+1,1 + + m p,1 (p 1n = d m j,1 + m j,ν. (6.2 P m 0,1 m 0,1 d + 2 d (6.3 λ j,ν + λ 0,1 / {0, 1,..., m 0,1 m 0,1 d + 2}. Riemann Fuchs m = ( m j,ν j=0,...,p ν=1,...,n j P (n p+1 d λ j,ν m j,1 d (j = 1,..., p m P (n d p+1 λ j,ν m j,ν = m j,ν δ ν,1 d (j = 0,..., p, ν = 1,..., n j, 2 λ 0,1 (j = 0, ν = 1 λ λ j,ν λ 0,1 + 1 (j = 0, ν > 1 j,ν =. 0 (j > 0, ν = 1 λ j,ν + λ 0,1 1 (j > 0, ν > 1 idx m = idx m, {λ m } = {λ m }

60 Proof. idx m idx m = n p j m j,ν λ j,ν j=0 ν=1 j=0 ν=1 p m 2 j,1 (p 1n 2 j=0 = 2d ( = d 2 p (m j,1 d 2 + (p 1(n d 2 j=0 p m j,1 (p + 1d 2 2(p 1nd + (p 1d 2 j=0 p j=0 m j,1 2d 2(p 1n = 0. n p j ( m j,νλ j,ν = m 0,1 (µ + 1 (m 0,1 d(1 µ + µ n m 0,1 j=0 p (n m j,1 ( p = m j,1 d (p 1n µ m 0,1 d (m 0,1 d = d. 6.2 Fuchs P Fuchs (5.10 (5.9 Riemann {λ m } n Fuchs 0 < n < n n n m = ( m j,ν 0 j p 1 ν n j P m j,ν 0 m j,ν m j,ν (0 j p, 1 ν n j, m p j,ν 1 ( λj,ν + i (p 1n(n 1 / {0, 1, 2,... }. 2 j=0 i=0 Proof. P u = 0 n F n F Fuchs P v = 0 P Fuchs c 0,..., c p c 1,..., c q P c j {λ j,ν i = 0,..., m j,ν 1, ν = 1,..., n j } ii P c j n m j,1,..., m j,n j {λ j,ν + k j,ν,i 1 i m j,1, ν = 1,..., n j } k j,ν,i 0 k j,ν,1 < k j,ν,2 < < k j,ν,m j,ν < m j,ν c j 0 k j,1 < k j,2 < < k j,n k j,i {k j,i 1 i n } P Fuchs (6.8

61 58 6 Fuchs {λ m } m = m + m, 0 < ord m < ord m m, m P {λ m } / {0, 1, 2,...} (6.9 Riemann {λ m } Fuchs k m P (n p+1 i λ j,ν (j = 0,..., p, ν = 1,..., n j (Fuchs Riemann {λ m } Fuchs ii idx m < 0 i λ j,ν km Riemann {λ km } Fuchs Proof. i m m = m + m, 0 < ord m < ord m {λ m } {0, 1, 2,...} {λ m } = 0 λ j,ν {λ m } Z l m = lm m m ii km = m + m, 0 < ord m < ord m {λ m } {0, 1, 2,...} l < k m = lm {λ lm } = n p j lm j,ν λ j,ν ord lm + l2 2 idx m j=0 ν=1 = l ( ord m k idx m l 2 l ord m idx m l(l k = idx m > 0 2 {λ m } > 0 m P idx m < m Fuchs generic P RAd ( µ (Tsai [Ts, Corollary 5.5]. Q W [x] C c 1,..., c p *2 x = c j λ j,ν {[0] (mj,1, [λ j,2 ] (mj,2,..., [λ j,nj ] (mj,nj } / Z Q W (x Q RQ W [x] W [x]rq *2 Tsai x =

62 Laplace Q W [x] Q = m 0 i= k x i q i (ϑ q i (s m 0, k, q m0 0, k > 0 x = Q Q x = {[λ 1 ] (m1,..., [λ n ] (mn } L(Q x = 0 {[0] (m0, [λ 1 1] (m1,..., [λ n 1] (mn } Proof. i 0 L ( x i q i (ϑ = ( i q i ( ϑ 1 r j (x (j = 1,..., k m 0 L(Q = ( i q i ( ϑ 1 + i=0 k x j r j (ϑ Fuchs P u = 0 Riemann (5.9 ( λ j,ν {λ m } = 0 0 j p 1 ν n j (λ j,ν 0 j p 1 ν n j C(λ C(λ W [x; λ] = W [x] C C(λ = { Q = n a i (x a i (x C(λ[x] } i=0 C W [x; λ] W (x; λ m j,1 = 0 (j = 1,..., p { λ j,1 = 0 (j = 1,..., p, { } (6.10 λ m = 0 RAd ( µ P, W (x; λ (6.10 generic P u = 0 µ = λ 0,1 1 ord P > 1 RAd ( µ RP

63 60 6 Fuchs Proof RP W [x; λ] Laplace LRP LRP / C(λ[x] LRP W (x; λ LRP C(λ[x] f(x C(λ[x] RP = f( f(x ord P > 1 L R P W (x; λ RAd (x µ RAd(x µ L R P W (x; λ RAd(x µ L R P x = 0 { [ µ]( p n j ν=2 m j,ν n, [0] } (m 0,1, [λ 0,2 1 µ] (m0,2,..., [λ 0,n0 1 µ] (m0,n RAd (x µ L R P W [x; λ] RAd ( µ P = L 1 RAd (x µ L R P C(λ[x] W (x; λ C(λ[x] P α C(λ[x] f(x C(λ[x] RAd (x µ L R P = + α, L R P = f(x(ϑ + αx µ R P = f( ( ϑ + α µ 1 P ord P > ord P = Katz RAd ( (x c τ RAd( µ Fuchs RAd ((x c τ RAd ( µ Fuchs Schlesinger *3 N. Katz middle convolution [Kz] *4, [DR] P u = 0 (P W [x] Riemann (5.9 Fuchs P = i,j p i,jx i j p i,j P Gauss x(1 x d2 u dx + ( γ (α + β + 1x du dx αβu = 0 *3 c 1,..., c p, du dx = p A j u u n A x c j j n *4 [Kz]

64 6.3 Katz 61 Riemann α 1 γ γ α β β α, β, γ *5 Gauss Heun Heun d 2 ( u γ dx 2 + x + δ x 1 + ɛ du x t dx + αβx q u = 0 (α + β + 1 = γ + δ + ɛ x(x 1(x t Riemann 0 1 t α 1 γ 1 δ 1 ɛ β α, β, γ, δ, ɛ, q q q q P u = 0 P P = n i=0 a i(x i a i (x = b i (x p (x c j i, b 0 (x = 1 deg b i (x (p 1n + i pi = (p 1(n i P c 0 =, c 1,..., c p Fuchs P n 1 ( (pn + p n + 1n (p 1(n i + 1 = 2 i=0 x = c j [λ j,ν ] (mj,ν (x c j m j,ν Ad ( (x c j λ j,ν P W [x] (x c j k P W [x] ( l a i (c j = 0 (0 l k 1, 0 i n *5

65 62 6 Fuchs ( l b i (c j = 0 (0 l k 1 i, 0 i k 1 b i (x (m j,ν + 1m j,ν 2 Fuchs n p j j=0 ν=1 m j,ν (m j,ν P (pn + p n + 1n 2 = (pn + p n + 1n 2 = 1 2( (p 1n 2 n p j j=0 ν=1 n p j j=0 ν=1 1 m j,ν (m j,ν m 2 j,ν (p + 1 n n p j m 2 j,ν + 2 = Pidx m j=0 Pidx m m Pidx m = 0 idx m = 2 m rigid rigid P u = 0 Riemann (5.9 Fuchs λ j,ν Fuchs P W [x; λ] *6 λ j,ν Fuchs generic P u = 0 ord P > 1 idx m 2 W (x; λ λ j,ν generic P idx m = 2 RAd ( (x c j τ RAd ( µ P Proof. j = 0,..., p m j,ν m j,1 = max{m j,ν ν = 1,..., n j } *6 λ j,ν Fuchs P [O6, Theorem 8.13] [O6, Example 7.5]

66 6.4 Riemann 63 p RAd ((x c j λ j,1 P Riemann x = c 0 c 1 c p [λ 0,1] (m0,1 [λ 1,1] (m1,1 [λ p,1] (mp,1.... [λ 0,n 0 ] (m0,n0 [λ 1,n 1 ] (m1,n1 [λ p,n p ] (mp,n p λ j,1 = 0 (j = 1,..., p λ j,ν Fuchs P λ j,ν generic µ = λ 0,1 1 RAd ( µ n d d = p j=0 m j,1 (p 1n addition generic n d 1 *7 ( p j=0 m j,1 (p 1n n = idx m + n p j (m j,1 m j,ν m j,ν. j=0 ν=1 d m j,1 2 0 idx m > 0 > 0 d > 0 0 < n d < n. ord m = 1 rigidity idx m = Riemann Riemann n P = a n (x n + a n 1 (x n a 0 (x n Fuchs Riemann (5.9 m j,ν 0 l = (l 1,..., l p (1 l j n j #{j m j,lj n, 0 j p} 2 *7 0

67 64 6 Fuchs d l (m := m 0,l0 + + m p,lp (p 1 ord m, ( p l P := Ad (x c j λ j,l j p (x c j m j,l j d l (m m 0,l 0 Ad( 1 λ 0,l0 λ p,l p m 0,l 0 d l (m a n (x 1 p ( p (x c j n m j,l j Ad (x c j λ j,l j P λ j,ν P l P W [x] m { λ m } l m := ( m j,ν 0 j p, 1 ν n j Riemann {λ m } m j,ν = m j,ν δ lj,νd l (m λ 0,ν 2µ l (j = 0, ν = l 0 l {λ m } := {λ m }, λ 0,ν µ l (j = 0, ν l 0 λ j,ν = λ j,ν (1 j p, ν = l j λ j,ν + µ l (1 j p, ν l j p µ l = λ j,lj 1. j=0 { } l λm l P Riemann m j,lj d l (m (j = 0,..., p l m well-defined m := (1,1,... m, l max (m l max (m j = min { ν m j,ν = max{m j,1, m j,2,...} } j max m := lmax (mm, p d max (m := max{m j,1, m j,2,..., m j,nj } (p 1 ord m j=0

68 6.4 Riemann idx m = idx l m, ord max m = ord m d max (m idx m > 0 d max (m > m P sm j m j,ν m P (n p+1 m n = ord m m P sm max m well-defined Proof. #{j m j,1 < n} < 2 max m well-defined P u = 0 m m j,1 = n c j [τ j ] (n RAd ( (x c j τ j cj P (5.15 (5.16 P = n m #{j m j,ν < n} m max m , 411, 42, =3 111, 111, =1 11, 11, , 1, 1 ( , 211, =1 111, 111, =0 111, 111, 111 ( , 211, 211, =1 111, 111, 111, = 1 211, 211, 211, 31 ( , 3311, =3 311, 311, =2 (idx m = , 3 d max (m 0 m m = ( m j,ν 0 j p 1 ν n j K P (n p+1 λ j,ν C m ord m > ord max m > > ord K maxm, s K maxm d max ( K maxm 0 k = 0,..., K m(k P p+1, l(k Z, λ(k j,ν C m(0 = m, m(k = max m(k 1,

69 66 6 Fuchs l(k = l max ( m(k, d(k = dmax ( m(k, {λ(k m(k } = k max{λ m } i rigid 2 6 2:11,11,11 3:111,111,21 3:21,21,21,21 4:1111,1111,31 4:1111,211,22 4:211,211,211 4:211,22,31,31 4:22,22,22,31 4:31,31,31,31,31 5:11111,11111,41 5:11111,221,32 5:2111,2111,32 5:2111,221,311 5:221,221,221 5:221,221,41,41 5:221,32,32,41 5:311,311,32,41 5:32,32,32,32 5:32,32,41,41,41 5:41,41,41,41,41,41 6:111111,111111,51 6:111111,222,42 6:111111,321,33 6:21111,222,33 6:21111,3111,33 6:2211,2211,411 6:222,222,321 6:222,3111,321 6:3111,3111,321 6:2211,33,42,51 6:222,33,411,51 6:321,321,42,51 6:33,33,33,42 6:33,33,411,42 6:33,411,411,42 6:411,411,411,42 6:33,42,42,51,51 6:321,33,51,51,51 6:411,42,42,51,51 6:51,51,51,51,51,51,51 ii rigidity 0 m P d max (m 0 *8 idx m = 0 ( :11,11,11,11 3:111,111,111 4:22,1111,1111 6:33,222, idx m = 2 ([O2, Proposition 8.4] 2:11,11,11,11,11 3:111,111,21,21 4:211,22,22,22 4:1111,22,22,31 4:1111,1111,211 5:11111,11111,32 5:11111,221,221 6:111111,2211,33 6:2211,222,222 8:22211,2222,44 8: ,332,44 10:22222,3331,55 12: ,444,66 idx m = 4 2:11,11,11,11,11,11 3:111,21,21,21,21 4:22,22,22,31,31 3:111,111,111,21 4:1111,22,22,22 4:1111,1111,31,31 4:211,211,22,22 4:1111,211,22,31 6:321,33,33,33 6:222,222,33,51 4:1111,1111,1111 5:11111,11111,311 5:11111,2111,221 6:111111,222,321 6:111111,21111,33 6:21111,222,222 6:111111,111111,42 6:222,33,33,42 6:111111,33,33,51 6:2211,2211,222 7: ,2221,43 7: ,331,331 7:2221,2221,331 8: ,3311,44 8:221111,2222,44 8:22211,22211,44 9:3321,333,333 9: ,333,54 9:22221,333,441 10: ,442,55 10:22222,3322,55 10:222211,3331,55 12: ,444,66 12:33321,3333,66 14: ,554,77 18: ,666,99 *8 rigidity d max (m 0 m P [O2, Proposition 8.1] m P p+1 ord m 3 idx m + 6, p 1 idx m + 3 [O6, Proposition 9.13] 2

70 67 7 Fuchs Riemann n n Fuchs Riemann Fuchs n Fuchs Deligne-Simpson quiver * m = (m j,ν 0 j p P (n p+1 1 ν n j m j,1 m j,2 m j,nj > 0, n > m 0,1 m 1,1 m p,1 > 0, (7.1 m 0,1 + + m p,1 (p 1n (7.2 n j m j,i = max{m j,ν i, 0} ν=1 ( : i= *1 Deligne-Simpson W. Crawley-Boevey [CB] P 1 (C/{c 0,..., c p } Deligne-Simpson (cf. [Ha2], [Ko2]

71 68 7 ν = 2, 3,..., n 1 #{(j, i Z 2 i 0, 0 j p, m j,i n ν} (p 1(ν (7.3 m k 2 (k, k ; k, k ; k, k ; k, k, (k, k, k ; k, k, k ; k, k, k, (2k, 2k ; k, k, k, k ; k, k, k, k, (3k, 3k ; 2k, 2k, 2k ; k, k, k, k, k, k. (7.4 Proof. n j φ j (t = max{m j,ν t, 0}, φj (t = n ( 1 t m j,1 ν=1 0 < t < m j,1 n n n j 0 1 m j,1 t φ j (t = φ j (t φ j (t < φ j (t (n = m j,1 n j, (, ν = 2,..., n 1 p #{i Z 0 φ j (i n ν} = j=0 p φ 1 j j=0 (n ν + 1 p ( φ 1 j (n ν + 1 j=0 p ( φ 1 j (n ν + 1 = p j=0 j=0 (νm j,1 (p 1ν + (p + 1 = (p 1(ν r r 2 ν n 1 ν (7.3 n + 1 φ 1 j (n ν Z (j = 0,..., p, n = m j,1 n j (j = 0,..., p, (p 1n = m 0,1 + + m p,1, (7.5 n = m j,1 n j m j,1 = = m j,nj = n n j p 1 = 1 n n p p+1 2 p = 2, 3 p = 3 n 0 = n 1 = n 2 = n 3 = 2 p = 2 1 = 1 n n n 2 p = 2 {n 0, n 1, n 2 } {3, 3, 3} {2, 4, 4} {2, 3, 6} (7.4 k 1 φ 1 1 j (n ν = φ j (n ν = νm j,1 n = ν n j Z (j = 0,..., p