Keiji Matsumoto (Hokkaido Univ.) Jan. 08, ,

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1 Keiji Mtsumoto (Hokkido Univ.) Jn. 08, ,

2 . > b > 0 { n }, {b n } ( 0, b 0 ) = (, b), ( n+, b n+ ) = ( n + b n, n b n ). { n }, {b n } lim n n = lim b n n b M(, b) Theorem (C.F. Guss 799 ) Mple M(, b) = F (,, ; ( b ) ), F (α, β, γ; z) Guss F (α, β, γ; z) = n=0 (α) n (β) n (γ) n () n z n, z <, (α) n = Γ (α+n) Γ (α), γ 0,,,...

3 . R + (R + )k m 0 k () m 0 () min(x) m 0 (x) mx(x), min(x) < mx(x) min(x) < m 0 (x) < mx(x) () m 0 (t x) = t m 0 (x) for t R + x = (x,..., x k ) (R + )k k k m,..., m k m m : (R + )k x m(x) = (m (x),..., m k (x)) (R + )k. (R + )k k {[n]} n N = {([n],..., [n] k )} n N m n [n] = m n () (n N)

4 Theorem ( ) k {[n]} n N k {[n] },..., {[n] k } m () (R + )k k Theorem ( ) m µ(m(x)) = µ(x), µ(t,..., t) = t for t R + µ Proof. k µ µ() = µ(m()) = µ(m ()) = = µ(m n ()) (n ) µ(m ()) = µ(m (),..., m ()) = m (). Remrk m i m

5 Exmple m, m, m m (x, x, x ) = x + x + x, m (x, x, x ) = x x + x x + x x, x + x + x m (x, x, x ) = x x x x x + x x + x x. m x (m (x), m (x), m (x)) m (x)m (x)m (x) x + x + x xx + x x + x x x + x + x x x x m µ(t, t, t) = t x x x x x + x x + x x = x x x m (x, x, x ) = x x x 4

6 . Guss Fct ( Guss ) ( + z) α F (α, α β +, β + ; z ) = F (α, β, β; 4z ( + z) ), z 0 z = 0 ( + z) α Proof of Theorem. Fct b = z +z, α = β = ( + b)/ F (,, ; ( b +b ) ) = F (,, ; ( b ) ) /F (,, ; ( b ) ) m m m = (m, m ) F (,,; ( b ) ) b = M(, b) 5

7 4. Gourst s formuls Guss 88 Gourst [G] E.M. Gourst, Sur l Éqution Différentielle Linéire qui Admet pour Intégrle l Série Hypergéométrique, Ann. Sci. l Ecole Normle Sup. () 0 (88), 4. pdf file p.7 p.7 40 Crlson [C] 6

8 (α, β, γ) { } γ, γ α β, α β F (,,, z) {,, }, b b/ m, m m = (m, m ) m n (, b) {( n, b n )} m (, b) type (M) { n }, {b n } b n b n+ n+ n or b n b n+ n+ n ; type (A) b 0 b b n n+ b n+ n b 0. 7

9 {,, } No. m (, b) m (, b) type m (, b) Q() b b( + b) (M) /F (,, ; b ) Q() ( + b) ( b (M) /F,, ; ) b ( (+b) +b Q() (M) /F,, ; ) ) ( b 4 Q(4) b +b b Q(5) b +b 4 +b ( b (M) /F,, ; ) ) ( b = b +b (M) /F (,, ; b ) = b 8

10 {, 4, 4} No. m (, b) m (, b) type m (, b) Q(6) b( + b) ( b (A) /F, 4, 5 4 ; ) b Q(7) b ( + b) (A) /F (,, 5 4 ; b ) b(+b) +b Q(8) (A) /F ( 4, 4, 5 4 ; ) ) ( b +b (+b) Q(9) (A) /F ( 4,, 5 4 ; ) ) ( b 4 Q(0) b +b b 4 +b b (A) /F ( 4, 4, 4 ; ) ) ( b = b ( 4 Q() +b b 4 b +b b (A) /F, 4, 4 ; ) ) ( b = b 9

11 [G] { } γ, γ α β, = {,, 6} α β b/, b (ξ, ξ ) b ξ + ξ =, ξ ξ = b, {ξ, ξ } = { ± b }, π 6 < rg(ξ i ) < π 6 X = ξ + ξ, X = ξ + ξ ξ + ξ, X = (i =, ) ξ ξ ξ + ξ, 0

12 (η, η ) b η η =, Y = η + η, Y = η + η = b, {η, η } = {b ± η + η η + η, Y = π 6 < rg(η i ) < π, (i =, ), 6 η b }, η = R + η η η + η,

13 No. m (, b) m (, b) type m (, b) C() b X b ( X (A) /F,, 7 6 ; ( ) ) b C() X X X (A) /F ( 6,, 7 6 ; ( b ) ) ( C() X X X X (A) /F,, ; ( ) ) b C(4) b X X b ( X X (A) /F 5 6,, ; ( ) ) b C(5) b X b ( X (A) /F,, ; ( ) ) b = b

14 No. m (, b) m (, b) type m (, b) C(6) Y ( Y (A) /F 6,, 7 6 ; ( ) ) b C(7) Y Y Y (A) / [ b C(8) Y Y Y Y (A) /F F ( (,, 7 6 ; ( b ) )], 5 6, ; ( b ) ) C(9) Y Y Y Y (A) /F,, ; ( ) ) b C(0) Y Y (A) /F ( 6,, ; ( ) ) b = b (

15 5. Borwein Borwein [BB] Theorem 4 F (,, ; x ) = F ( 4, 4, ; x ) = + x F (,, ; ( x + x ) ), F ( + x 4, 4, ; ( x + x ) ). Theorem 5 m (, b) m (, b) m (, b) +b +b 4 b( +b+b b( +b (,, ; ( b ) ) ) /F ( ) /F 4, 4, ; ( ) ) b 4

16 6. Luricell F D Luricell k- F D F D (α, β, γ; z) = n,...,n k 0 (α, k j= n j ) k j= (β j, n j ) (γ, k j= n j ) k j= (, n j ) k j= z n j j, z = (z,..., z k ) z j < (j =,..., k) β = (β,..., β k ), γ 0,,,... k = F D (α, β, γ; z) Guss F (α, β, γ; z) k = Appell F (α, β, β, γ; z, z ) F D (α, β, γ; z) = Γ (γ) Γ (α)γ (γ α) 0 tα ( t) γ α m j= ( z j t) β j dt t( t). 5

17 Fct F D (α, β, γ; z) dω ˆf (z) = Ω ˆf (z) Ω ˆf (z) d ˆf = Ω ˆf (z) ˆf, Ω ˆf (z) = i<j k+ A ij d log(z i z j ), ˆf = t (f 0, f,..., f k ), f 0 = F D (α, β, γ; z), f i = z i f 0 z i ( i k), z k+ = 0, z k+ =, (k + ) (k + )- A ij A ij = 0-th i-th j-th 0-th i-th β j β i j-th β j β i ( i < j k), 6

18 A i,k+ = A i,k+ = 0-th i-th 0-th β O. O β i j i i-th γ + β j j k β i+ O. O β k ( i k), 0-th i-th 0-th O O i-th αβ i β i β i γ α β i β i β i ( i k). O O

19 7. F D Theorem 6 F D ( ) + z + z p F D ( p, p + 6, p + 6, p + ; z, z ) = F D ( p, p + 6, p + 6, p ; z, z ), z = (z, z ) (, ) ( +z +z ) p (, ) z = ( + ωz + ω z + z + z ), z = ( + ω z + ωz + z + z ), ω = + 7

20 p = ( ) ( ) Theorem 7 F D (z z ) p ( ) z + z p ( + p F D 4, + p 4, + p 4, + p ) ; z 4, z ( = F D p, + p 4, + p 4, + p ; z ( + z ), z ) ( + z ), 4 z + z z + z z = (z, z ) (, ) (z z ) p (, ) ( ) z +z p 8

21 Theorem 8 F D ( +z +z +z 4 ) p FD ( p 4, p+, p+, p+, p+ ; z, z, z ) = F D ( p 4, p +, p +, p +, p ; z, z, z ), (z, z, z ) (,, ) ( +z +z +z 4 ) p/ (,, ) z = ( z z + z + z + z + z ) = 4( + z )(z + z ) ( + z + z + z ), z = ( z + z z + z + z + z ) = 4( + z )(z + z ) ( + z + z + z ), z = ( + z z z + z + z + z ) = 4( + z )(z + z ) ( + z + z + z ) Theorems 6,7,8 9

22 8. m, m, m m = (m, m, m ) m (x, x, x ) = x + x + x, m (x, x, x ) = m (x, x, x ) l (x, x, x ), m (x, x, x ) = m (x, x, x ) l (x, x, x ), l (x) = x +ωx +ω x, l (x) = x +ω x +ωx > b > c > 0 ( n, b n, c n ) m n (, b, c) Theorem 9 (Koike-Shig) { n }, {b n }, {c n } m (, b, c) F D m (, b, c) = F D (,,, ; ( b ), ( c ) ). 0

23 m, m, m m = (m, m, m ) x ( x + x ) m (x, x, x ) =, x ( x + x ) m (x, x, x ) =, x ( x + x ) m (x, x, x ) =. > b > c > 0 ( n, b n, c n ) m n (, b, c) Theorem 0 { n }, {b n }, {c n } m F D m (, b, c) = F D (,,, ; b, c ). Mple (, b, c)

24 m,..., m 4 m = (m,..., m 4 ) m (x) = x + x + x + x 4 (x + x 4 )(x + x ), m (x) =, 4 (x + x )(x + x 4 ) (x + x )(x + x 4 ) m (x) =, m 4 (x) =. > b > c > d > 0 ( n, b n, c n, d n ) m n (, b, c, d) Theorem { n }, {b n }, {c n }, {d n } m (, b, c, d) F D m (, b, c, d) = Mple F D ( 4, 4, 4, 4, ; ( b ), ( c ), ( d ) ).

25 Remrk 876 C.W. Borchrdt m (x) = x + x + x + x 4 x x 4 + x x, m (x) =, 4 x x + x x 4 x x + x x 4 m (x) =, m 4 (x) =.

26 References [B] C.W. Borchrdt, Über ds rithmetisch-geometrische Mittel us vier Elementen, Berl. Montsber, 5 (876), 6-6. BB] J.M. Borwein nd P.B. Borwein, A cubic counterprt of Jcobi s identity nd the AGM, Trns. Amer. Mth. Soc., () (99), BB] J.M. Borwein nd P.B. Borwein, Pi nd the AGM (Reprint of the 987 originl), A Wiley-Interscience Publiction. John Wiley & Sons, Inc., New York, Toronto,

27 [C] B.C. Crlson, Algorithms involving rithmetic nd geometric mens, MAA Monthly, 78(5) (97), [E] A. Erdeélyi, Higher Trnscendentl Functions, volume I, Robert E. Krieger Publishing Co., Inc., Melbourne, Florid, 98. [G] E.M. Gourst, Sur l Éqution Différentielle Linéire qui Admet pour Intégrle l Série Hypergéométrique, Ann. Sci. l Ecole Normle Sup. () 0 (88), 4. KM] R. Httori, T. Kto nd K. Mtsumoto, Men itertions derived from trnsformtion formuls for the hypergeometric function, preprint

28 KSY] K. Iwski, H. Kimur, S. Shimomur nd M. Yoshid, From Guss to Pinlevé, Vieweg, Brunschweig, Wiesbden, 99. [KM] T. Kto nd K. Mtsumoto, The common limit of qudruple sequence nd the hypergeometric function F D of three vribles, preprint, 007. [KS] K. Koike nd H. Shig, Isogeny formuls for the Picrd modulr form nd three terms rithmetic geometric men, J. Number Theory, 4 (007), 4. [KS] K. Koike nd H. Shig, Extended Guss AGM nd corresponding Picrd modulr forms, J. Number Theory, 8 (008),

29 [MM] D.V. Mnn nd V.H. Moll, Lnden survey, Probbility, Geometry nd Integrble Systems, 87-9, MSRI Publictions 55, Cmbridge University Press, Cmbridge, 008. [MO] K. Mtsumoto nd K. Ohr, Some trnsformtion formuls for Luricell s hypergeometric functions F D, to pper in Funkcil. Ekvc. [MS] K. Mtsumoto nd H. Shig, A vrint of Jcobi type formul for Picrd curves, preprint, 008. [MT] K. Mtsumoto nd T. Tersom, Arithmetic-geometric mens for hyperelliptic curves nd Clbi-Yu vrieties, to pper in Internt. J. of Mth. 7

30 [M] K. Mtsumoto, A trnsformtion formul for Appell s hypergeometric function F nd common limits of triple sequences, preprint, 008. [Me] J.F. Mestre, Moyenne de Borchrdt et intégrles elliptiques, C. R. Acd. Sci. Pris, (99), [O] K. Ohr, yng pckge for computtion in the ring of differentil-difference opertors, [U] H. Umemur, - -, University of Tokyo Press, Tokyo,

第86回日本感染症学会総会学術集会後抄録（I）

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