1 Fuchs. Fuchs. Gauss (1.1) (α) n := α(α + 1) (α + n 1) Bessel Riemann. [MUI], [WW] Gauss. (1.2) x(1 x) d2 u dx 2 + ( γ (α + β + 1)x ) du

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1 Fuchs Kac-Moody root 1 Gauss 1. Fuchs (1.1) F (α, β, γ; x) = n=0 (α) n (β n ) x n (γ) n n!, (α) n := α(α + 1) (α + n 1) Bessel Riemann [MUI], [WW] Gauss (1.2) x(1 x) d2 u dx 2 + ( γ (α + β + 1)x ) du dx αβu = 0 1 Fuchs Fuchs n n Riemann P 1 (C) {c 0,..., c p } X 1. X U U O(U) n n F(U), U F(U) F (1.3) X U V : = F(V ) = F(U) V 2. F(U) 0 F(U) u(x) C, N, ɛ (1.4) u(x) < C x N (0 < x < ɛ) U {x C ; e iθ x / R + } θ R 3. n n 1 Wronskian X F(U) Wronskian X c 0 =

2 F P P u = 0 (1.5) ( p P = (x c j ) n) d n n 1 dx n + ( p a k (x) (x c j ) k) d k dx k, k=0 a k (x) C[x], deg a k (x) (n k)(p 1). Fuchs 2. F(U) P u = 0 u 1 (x),...,u n (x) u ν (x) x λ ν log k ν x. (λ ν, k ν ) (ν = 1,..., n). {λ ν ; ν = 1,..., n} {0, 1,..., n 1} Gauss, 0, 1, {α, β}, {0, 1 γ}, {0, γ α β} Riemann scheme (2.1) x = x = 0 x = 1 α 0 0 ; x β 1 γ γ α β F ũ = (ũ 1,..., ũ n ) c j γ j M j M j (2.2) q γ 3 γ 0 γ γ 2 1 c 0 c 1 c 2 γ i γ j (ũ) = γ j (ũm i ) = γ j (ũ)m i = ũm j M i, M p M p 1 M 1 M 0 = I n. c j λ j,1,..., λ j,n M j e 2πiλ j,1,..., e 2πiλ j,n λ j,ν λ j,ν / Z (1 ν < ν n) M j Fuchs (α 1 ) ν (α n ) ν (2.3) nf n 1 (α 1,..., α n, β 1,..., β n 1 ; x) = (β 1 ) ν (β n 1 ) ν ν! xν, 0, 1 1 n 1 1 0, 1,..., n 2 2 ν=0

3 Jordan-Pochhammer, c j p + 1 p, c j j = 1,..., p) 0,..., p 2 p 1 λ, λ + 1,..., λ + p 1 λ x λ p c j {[λ j,1 ] (mj,1),..., [λ j,nj ] (mj,nj )} (2.4) {λ j,ν + k ; k = 0,..., m j,ν 1, ν = 1,..., n j } (2.5) λ j,ν λ j,ν / Z n = m j,1 + + m j,nj λ j,ν = λ j,1 (ν = 2,..., n j ) (2.4) Jordan n m j,1 + + m j,nj dual [Os5] Riemann scheme x = c 0 c 1 c p [λ 0,1 ] (m0,1 ) [λ 1,1 ] (m1,1 ) [λ p,1 ] (mp,1 ) (2.6) {λ m } :=..... ; x [λ 0,n0 ] (m0,n0 ) [λ 1,n1 ] (m1,n1 ) [λ p,np ] (mp,np ) p + 1 n (2.7) n = m j,1 + + m j,nj (j = 0,..., p) F P Jordan-Pochhammer Riemann scheme x = 0 1 (2.8) (2.9) 1 β 1 [0] (n 1) α 1.. ; x, n α ν = ν=1 n β ν, ν=1 1 β n 1 α n 1 0 β n α n x = c 1 c p [λ 0] (p 1) [0] (p 1) [0] (p 1) ; x, (p 1)λ 0 + λ 0 λ 1 λ p p λ j = p 1 j=0 1 n, n 11, 1 n p 11, p 11,, p 11 }{{} p+1 n = 2, p = 2 Gauss 11, 11, 11 3

4 m = {m j,ν } 0 j p 1 ν n j (2.10) m j,ν = nδ ν,1, n j = 1 n p + 1 n = ord m (j > p), m j,ν = 0 (ν > n j ) {λ m } Fuchs Riemann scheme (GRS) Fuchs (FC) n p j (2.11) {λ m } := m j,ν λ j,ν ord m + idx m = 0. (2.12) (2.13) j=0 ν=1 ( p idx(m, m ) := lim p j=0 ν=1 idx m := idx(m, m). m j,ν m j,ν (p 1) ord m ord m ),. m Fuchs (FC) generic λ j,ν GRS) {λ m } F Fuchs m irreduciblyrealizable Fuchs 3. Kac-Moody root (3.1) I := {0, (j, ν) ; j = 0, 1,..., ν = 1, 2,...}. α i (i I) h s i W Kac-Moody root (Π, W ) α i W Weyl (3.2) Π = {α i ; i I} = {α 0, α j,ν ; j = 0, 1, 2,..., ν = 1, 2,...}. (3.3) (3.4) I := I \ {0}, Π := Π \ {α 0 }, Q := α Π Zα Q + := α Π Z 0 α. (3.5) (α α) = 2 (α Π), (α 0 α j,ν ) = δ ν,1, 0 (i j or µ ν > 1), (α i,µ α j,ν ) = 1 (i = j and µ ν = 1). 4 α 0 α 0,1 α 0,2 α 1,1 α 1,2 α 2,1 α 2,2 α 3,1 α 3,2

5 (α α) 0 α h s α (3.6) (3.7) s α : h x x 2 (x α) (α α) α h, s i = s αi for i I. = re im + = Q + (3.8) (3.9) (3.10) re := W Π (), re + := re Q +, im + := W B () B := {β Q + ; supp β is connected and (β, α) 0 im := im + im, im := im +. ( α Π)}, w W, α Q (3.11) (3.12) (3.13) (w) + := re + w 1 re, L(w) := # (w) +, h(α) := n 0 + n j,ν for α = n 0 α 0 + n j,ν α j,ν Q. j 0 ν 1 j 0 ν 1 w = s i1 s i2 s ik i ν I w W minimal expression (3.14) (w) + = { α ik, s ik (α ik 1 ), s ik s ik 1 (α ik 2 ),..., s ik s i2 (α i1 ) }. α + wα B {α 0 } w W w (α) + := (w) + h (3.15) (3.16) (3.17) (3.18) (3.19) h := {Λ = λ i α i Cα i ; λ j,1 = 0 (j 1)}, i I i I Λ 0 := 1 2 α i=ν+1 (1 ν)α j,ν, j=0 ν=1 Λ j,ν := (i ν)α j,i (j = 0,..., p, ν = 0, 1, 2,...), Λ 0 := 2Λ 0 2Λ 0,0 = α 0 + (1 + ν)α 0,ν + (1 ν)α j,ν, ν=1 ν=1 Λ 0 j,k := Λ j,0 Λ k,0 = ν(α k,ν α j,ν ) (0 j < k). ν=1 (3.20) (3.21) (3.22) (Λ 0 α) = (Λ 0 j,k α) = 0 ( α Π), (Λ j,ν α j,ν ) = δ j,j δ ν,ν (j, j = 0, 1,..., ν, ν = 1, 2,...), (Λ j,0 α i ) = δ i,0 ( i Π). 5

6 4. reduction. u(x) (x c j ) λj u(x) (4.1) Ad ( (x c j ) λ j ) : x x, (x c j ) d dx (x c j) d dx λ j addition gauge fractional Euler (4.2) u(x) µ u(x) = 1 Γ(µ) (4.3) µ : d dx d dx, x c j (x s) µ 1 u(s)ds x d dx x d dx µ Weyl. Weyl P P u(x) = 0 u(x) u(x) v(x) v(x) Qv(x) = 0 Q Weyl. P (2.6) GRS (1.5) Fuchs addition Ad ( ) (x c j ) λ j l = (l0,..., l p ) Z p+1 0 Euler (4.4) l P := Ad ( p p (x c j ) λ j,l j ) (x c j ) m j,l j d l (m) m 0,l 0 Ad( 1 λ 0,l0 λ p,l p ) (p 1)n m 1,l 1 m p,l p a 1 n (x) n (x c j ) n m ( p j,l j Ad (x c j ) λ j,l j )P. d l (m) := m 0,l0 + + m p,lp (p 1) ord m. l P m = l m, GRS {λ m } = l{λ m } λ j,ν Fuchs generic (4.5) m j,ν = m j,ν δ ν,lj d l (m), λ j,ν = λ j,ν + (1 δ ν,lj 2δ j,0 )µ, µ = λ 0,l0 + + λ p,lp 1 (4.6) ord l m = ord m d l (m). m j,lj 0 l m ord l m d l (m) l l l max (m) (4.7) m j,lmax (m) j = max{m j,1,..., m j,nj } max m = lmax(m)(m) d max (m), max {λ m } max P m irreducibly realizable K reduction (4.8) ord m > ord max m > ord 2 maxm > > ord K maxm, ord K m = 1 or d max ( K maxm) 0. 6

7 {λ m } Kac-Moody Schlesinger [CB] (4.9) α l := α 0 + l j 1 j=0 ν=1 α m := ord m α 0 + Λ(λ) := Λ 0 + p j=0 ν=1 α j,ν re +, p j=0 ν=1 i=ν+1 m j,i α j,ν Q +, λ j,ν (Λ j,ν 1 Λ j,ν ) h := h /CΛ 0. { Pm : Fuchs with {λ m } } { } (Λ(λ), α m ) ; α m + l, addition W -action, +τλ 0 0,j { Pm : Fuchs with {λ m } } { (Λ(λ), α m ) ; α m + }. P m Kac-Moody root system α m m : rigid α re + : supp α α 0 m : monotone α Q + : (α β) 0 ( β Π ) m : realizable kα : k Z >0, α +, supp α α 0 m : irreducibly realizable α +, supp α α 0 indivisible or (α α) < 0 m : basic and monotone α Q + : (α β) 0 ( β Π) indivisible α + : (α α m ) = 1 ( α (m) + ) m: simply reducible and monotone (α β) 0 ( β Π ) (α α 0 ) > 0, α α 0, indivisible ord m n 0 : α = n 0 α 0 + i,ν n i,να i,ν idx(m, m ) (α m α m ) l s αl {λ m } (Λ(λ), m) {λ m } (Λ(λ) α m α m ) 4.1 ( [Os5]). m irreducibly realizable {λ m } GRS (1.5) P m (λ, g 1,..., g N ) Fuchs λ j,ν generic m simply reducible {λ m } GRS Fuchs (g 1,..., g N ) C N g 1,..., g N (4.10) N = idx m. P m (λ, g) (x, λ, g) g i 1 g i x ν dj dx j 7

8 irreducibly realizable m idx m = 2 rigid m indivisible {m j,ν } 1 basicindivisible d max (m) 0 monotone m j,1 m j,2 m j,3 m idx m (cf. [Os3]). idx m = 0 affine root D 4, Ẽ6, Ẽ7, Ẽ8 4 cf. [Ko]idx m = 2 13 [Os3] irreducible realizable m reduction (4.8) (4.11) ord i maxm = ord i 1 maxm 1 (i = 1,..., K) m simply reducible simply reducible non-rigid m idx m (cf. [Os5]) rigid simply reducible 21111, 222, 33 Simpson in [Si] (cf. [MWZ]) order type name partitions n H n hypergeometric family 1 n, 1 n, n 11 2m EO 2m even family 1 2m, mm 11, mm 2m + 1 EO 2m+1 odd family 1 2m+1, mm1, m + 1m 6 X 6 extra case , 222, 42 rigid m ord m [Os5] or reduction (5.1) P m (λ)u = (). m irreducible realizable (5.1) (5.2) (Λ(λ) α) / Z ( α (m) + ). (5.3) (m) + := (α m ) +. c 0 = 0, c 1 = 1m 0,n0 = 1 x = 0 u 0,n0 x λ 0,n 0 1 x = 1 u 1,n1 (1 x) λ 1,n 1 c(λ 0,n0 λ 1,n1 ). 5.2 (). l 0 n 0, l 1 n 1 l Z p 1 {λ m } = l{λ m } P m (λ )v = 0 c (λ 0,n 0 λ 1,n 1 ) (5.4) c (λ 0,n 0 λ 1,n 1 ) Γ(λ 0,n 0 λ 0,1 + 1)Γ(λ 1,1 λ 1,n 1 ) = c(λ 0,n0 λ 1,n1 ) Γ(λ 0,n0 λ 0,1 + 1)Γ(λ 1,1 λ 1,n1 ). 8

9 m rigid m 1,n1 = m 2,n2 = 1 c(λ 1,n1 λ 2,n2 ) cf. (4.8) (4.8) {λ(k) m(k) } = k max{λ m } k = 0,..., Kλ(k) j,max = λ(k) j,lmax (m(k)) j 5.3 (). m rigid m 1,n1 = 1, c 0 =, c 1 = 1 x = 0 λ 1,n1 u(x) u(x) u(x) x λ1,n 1 (5.5) (5.6) u(x) := K 1 k=0 s0 0 K 1 k=0 Γ ( λ(k) 1,n1 λ(k) 1,max + 1 ) Γ ( λ(k) 1,n1 λ(k) 1,max + µ(k) + 1 ) Γ ( µ(k) ) sk 1 K 1 0 s λ(k) 1,n 1 K (s k s k+1 ) µ(k) 1 k=0 ( ( sk s k+1 ) λ(k)1,max p j=2 p j=2 p ( = x λ 1,n 1 1 x ) λ(0)j,max c j K 1 i=0 K j=2 ( 1 c 1 j s k ) ) λ(k)j,max 1 c 1 j s k+1 ( 1 s ) K λ(k)j,max s0 dsk ds 1 c j =x (ν j,k ) 2 j p Z (p 1)K 0 1 k K ( λ(i)1,n1 λ(i) 1,max + 1 ) p s=2 K t=i+1 ν s,t ( λ(i)1,n1 λ(i) 1,max + µ(i) + 1 ) p i=1 s=2 ( ) p λ(i 1)s,max λ(i) s,max ν s,i! ν s,i s=2 K t=i+1 ν s,t p s=2 ( x c s ) K i=1 ν s,i. 5.4 (). m rigid m j,nj = 1 j = 0, 1, 2, c 0 = (5.7) ɛ j,ν = δ j,1 δ ν,n1 δ j,2 δ ν,n2, ɛ j,ν = δ j,0 δ ν,n0 δ j,2 δ ν,n2 j = 0,..., p, ν = 1,..., n j. u λ (x) (5.1) x = c 1 λ j,ν generic (5.8) u λ (x) (x c 1 ) λ1,n 1 (5.9) K 1 u λ (x) = u λ+ɛ (x) + (c 1 c 2 ) ν=0 λ(ν + 1) 1,n1 λ(ν) 1,l(ν)1 + 1 λ(ν) 1,n1 λ(ν) 1,l(ν)1 + 1 u λ+ɛ (x). cf. [Os2], [Os6] Gauss (1.2) α, β, α γ, β γ / Z Gauss F (α, β, γ; 1) = Γ(γ)Γ(γ α β) Γ(γ α)γ(γ β) 9

10 F (α, β, γ; x) = 3 Γ(γ) Γ(α)Γ(γ α) x 0 x γ+1 (x s) α+γ 1 s α 1 (1 s) β ds F (α + 1, β, γ; x) F (α, β, γ; x) = βx F (α + 1, β + 1, γ + 1; x) γ References [CB] W. Crawley-Boevey, On matrices in prescribed conjugacy classes with no common invariant subspaces and sum zero, Duke Math. J. 118 (2003), [DG] M. Dettweiler and S. Reiter, An algorithm of Katz and its applications to the inverse Galois problems, J. Symbolic Comput. 30 (2000), [DG2], Middle convolution of Fuchsian systems and the construction of rigid differential systems. J. Algebra 318 (2007), [Ha] Y. Haraoka, Integral representations of solutions of differential equations free from accessory parameters, Adv. Math. 169 (2002), [Ha2], 7,, [HF] Y. Haraoka and G. M. Filipuk, Middle convolution and deformation for Fuchsian systems, J. Lond. Math. Soc. 76 (2007), [Kc] V. C. Kac, Infinite dimensional Lie algebras, Third Edition, Cambridge Univ. Press [Ka] N. M. Katz, Rigid Local Systems, Annals of Mathematics Studies 139, Princeton University Press, [Ko] V. P. Kostov, The Deligne-Simpson problem for zero index of rigidity, Perspective in Complex Analysis, Differential Geometry and Mathematical Physics, World Scientific 2001, [MWZ] P. Magyar, J. Weyman and A. Zelevinski, Multiple flag variety of finite type, Adv. in Math. 141 (1999), [MUI] III 1960 [Os1] T. Oshima, Heckman-Opdam hypergeometric functions and their specializations, Harmonische Analysis und Darstellungstheorie Topologischer Gruppen, Mathematisches Forschungsinstitut Oberwolfach, Report 49 (2007), [Os2], Okubo, a computer program for Katz/Yokoyama/Oshima algorithms on MS- Windows, ftp://akagi.ms.u-tokyo.ac.jp/pub/math/okubo/okubo.zip, [Os3], Classification of Fuchsian systems and their connection problem, arxiv: , preprint, 2008, 29pp. [Os4], Katz s middle convolution and Yokoyama s extending operation, arxiv: , 2008, 18pp. [Os5], Fractional calculus of Weyl algebra and Fuchsian differential equations, preprint, 176pp, [Os6], muldif.rr, a library of the calculation of differential operators for computer algebra Risa/Asir, ftp://akagi.ms.u-tokyo.ac.jp/pub/math/muldif/, [Si] C. T. Simpson, Products of Matrices, Canadian Math. Soc. Conference Proceedings 12, AMS, Providence RI (1991), [Si2], Katz s middle convolution algorithm, 53 pp, arxiv:math/ [WW] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Fourth Edition, 1927, Cambridge University Press. [Yo] T. Yokoyama, Construction of systems of differential equations of Okubo normal form with rigid monodromy, Math. Nachr. 279 (2006),

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