A three terms Arithmetic-Geometric mean Kenji Koike Faculty of Education University of Yamanashi Takeda 4-4-7,Kofu Yamanashi 400-850, Japan kkoike@yamanashi.ac.jp Hironori Shiga Inst. of Math. Physics Chiba University Yayoi cho, Inage ku Chiba 26 8522, Japan shiga@math.s.chiba-u.ac.jp june 04,2005 0 Introduction After the discovery of arithmetic geometric mean of Gauss in 796, there has been proposed several its variants generalizations. There are some interesting studies among them, like the cubic-agm of Borweins in two terms case [B-B]. Also it has been aimed to find a new three terms AGM, for example the trial of Richelot ([R], also [B-M]). But up to present time there was no nicely settled theory for it. In this article we show a new AGM of three terms that has an expression by the Appell hypergeometric function. Our AGM is related to the Picard modular form appeared in [S], this story will be published elsewhere. Definition Definition. Let a, b, c be real positive numbers satisfying a b c. Set with Ψ(a, b, c) = (α, β, γ) = ( a + b + c, A, B), (.) A = 6 (a2 b + b 2 c + c 2 a + ab 2 + bc 2 + ca 2 ) + 6 (a c)(a b)(b c) B = 6 (a2 b + b 2 c + c 2 a + ab 2 + bc 2 + ca 2 ) 6 (a c)(a b)(b c). Here we choose the areguments of β = A γ = B such that 0 arg A < π 6, 0 < β + γ. Remark. If we make a permutation of a, b, c, it causes only the difference of the choice of complex conjugates. Lemma. For a triple (α, β, γ) = Ψ(a, b, c), we find uniquely determined real triple (x, y, z) = T (α, β, γ) such that (α, β, γ) = (x, y, z) ω 2 ω, (ω = exp(2πi/)) ω ω 2 x > y, x > z.
We obtain the assertion by an easy direct calculation. For a general triple (a, b, c) of complex numbers (x, y, z) = T (a, b, c), we have x = α, x y = β, x z = γ with (α, β, γ) = Ψ(a, b, c), we don t mind the branches of β, γ. By using the above notations, we have Proposition. For a real triple (a, b, c) with a b c we can determine Ψ(Ψ(a, b, c)) = (a, b, c ) = (x, x y, x z ). (.2) Especially the twice composite of Ψ induces a real positive triple again. Lemma.2 We have Put β = s + it. Then β γ β γ, β α β α β 2 + βγ + γ 2 β γ 2 s 2 t 2 /. Because of our choice of the argument the first inequality holds. We have α β α γ = 27(x 2 + xy + y 2 )(x 2 + xz + z 2 )(y 2 + yz + z 2 ) α β α γ = 27(y 2 + yz + z 2 ). It holds { (x 2 + xy + y 2 ) (y 2 + yz + z 2 ) = (x z)(x + y + z) = α(x z) > 0 (x 2 + xz + z 2 ) (y 2 + yz + z 2 ) = (x y)(x + y + z) = α(x z) > 0. So we obtain the second inequality. Proposition.2 We have β α 2 = 8 ((a b) 2 + (b c) 2 + (a c) 2 ), We have β γ = (a c)(b c)(a b) β γ 4 (a c), α β (a c), α γ (a c). β α 2 = 6 ( a 2 a b + b 2 a c b c + c 2) = 6 2 ((a b)2 + (b c) 2 + (a c) 2 ). So we get the equalities. By using the equalities the above Lemma together with the inequalities (a b) 2 + (b c) 2 (a c) 2, (b c)(a b) 4 (a c)2, we obtain the required inequalities. 2
Proposition. We have b c < 0.982(a c), a b < 0.2255(a c), a c < 0.2255(a c). (.) [ Acoording to the above Proposition we have: So it holds b c b c = y z = β γ α β 2 (a c) 4 9 (a c)2. Hence y z 2 ( 2/9) ( 4/) (a c) < 0.982(a c), y z 2 ( 2/9) ( 4/) (a c) < 0.982(a c). y ω 2 z = α β (a c). b a b a = y ( ( y z + z ωy )) [ {2 ( 2/9) ( 4/) + ( /2) }] (a c). So we have By changing the roles of β γ we have a b < 0.2255(a c). a c < 0.2255(a c). By Proposition.2 Proposition. we have the following. Theorem. Let a, b, c be real positive numbers satisfying the condition a b c. Set Ψ n (a, b, c) = (a n, b n, c n )(= Ψ(a n, b n, c n )). Then there is a common limit M(a, b, c) = lim n a n = lim n b n = lim n c n. 2 Functional equation Appell s HGDE 2. Functional equation Let x, y be real positive numbers. By the limit process of Ψ(, x, y) = ( + x + y, β(, x, y), γ(, x, y)) = + x + y (, β + x + y, γ + x + y ) we have Set Proposition 2. We have M(, x, y) = + x + y β M(, + x + y, γ ). (2.) + x + y H(x, y) = M(, x, y). H(x, y) = + x + y H( β + x + y, γ + x + y ).
2.2 Appell s Hypergeometric function F According to Appell [App] we define the hypergeometric function F with parameters a, b, b, c (see also [Y]): Definition 2. F (a, b, b, c; x, y) = m,n 0 (a, m + n)(b, m)(b, n) x m y n, (2.2) (c, m + n)m!n!,where we use the conventional notation { λ(λ + ) (λ + k ), (k ) (λ, k) = 0, (k = 0) f = F (a, b, b, c; x, y) satisfies the following linear partial differential equation x( x)f xx + ( x)yf xy + (c (a + b + )x)f x byf y abf = 0 y( y)f yy + ( y)xf xy + (c (a + b + )y)f y b xf x ab f = 0 (x y)f xy b f x + bf y = 0. (2.) It has a three-dimensional space of solutions at a regular point, it has unique holomorphic solution up to constant at the origin. The third equation is derived from the first the second. Differential equation Proposition. We have. F (/, /, /, ; x, y ) = We use the abreviation set ( + x + y F /, /, /, ; F (x, y) = F (/, /, /, ; x, y), z(x, y) = F (/, /, /, ; x, y ). ( + ωx + ω 2 ) y, + x + y ( + ω 2 ) ) x + ωy + x + y We describe the differental equation for z(x, y) deform it to an equivalent system under a change of variables. At first we obtain the system δ = x ( x ) xx + x y ( y ) 2 xy x x + y ( y ) 2 y δ 2 = y ( y ) yy + y x ( x ) 2 xy y y + x ( x ) 2 x (.) δ = (x y ) xy y 2 x + x 2 y for z(x, y), where we use the convention x = x etc. Put X = +ωx+ω2 y +x+y, Y = +ω2 x+ωy +x+y, { P = + X + Y Q = + ω 2 X + ωy R = + ωx + ω 2 Y 4
Then we have D X = (ω X) X + (ω 2 Y ) Y D Y = (ω 2 X) X + (ω Y ) Y. ( + x + y)( + X + Y ) = x = +X+Y D X y = +X+Y D Y. We can rewrite the system (.) in terms of X, Y, D X, D Y : δ = 9Q (P Q )(D X D X D Y ) + Q 9R (P R )(D 2 X D Y D Y ) QD X + R (P R )D 2 Y δ 2 = 9R (P R )(D Y D Y D X ) + R 9Q 2 (P Q )(D Y D X D X ) RD Y + Q 2 (P Q )D X Namely, we have δ z = δ 2 z = δ z = 0 δ Z = δ 2Z = 0, where z = z(x, y) = Z(X, Y ) = Z. We obtain the system for Z = δ = 9Q (P Q )(D X D X D X + 2) + Q 9R 2 (P R )(D X D Y D X 2D Y + 2) Q(D X ) + R 2 (P R )(D Y ) δ 2 = 9QR (P R )(D Y D Y D Y + 2) + R 9Q (P Q )(D 2 Y D X D Y 2D X + 2) R(D Y ) + Q (P Q )(D 2 X ), Z +x+y +X+Y (= z(x, y)). Then we have δ Z(X, Y ) = δ 2Z(X, Y ) = 0 δ Z(X, Y ) = δ 2 Z(X, Y ) = 0. On the other h, we can rewrite the system (2.) for the function F (X, Y ) that is coming from the right h side. Namely we get δ = X ( X ) XX + Y X ( X ) 2 XY + X ( X ) 2 X Y Y δ 2 = Y ( Y ) Y Y + X Y 2 ( Y ) XY + Y 2 ( Y ) Y X X δ = X 2 Y 2 (Y X ) XY + X 2 X Y 2 Y. By direct calculation we get the following equality of the differential operators: QR P (Rδ + Qδ 2 ) = X( + 2X)(X Y 2 )δ + Y ( + 2Y )(Y X 2 )δ2 + XY (Y X)(XY )δ QR P (Rδ Qδ 2 ) = X(X Y 2 )δ + Y (Y X 2 )δ 2 + XY (X + Y )(XY )δ. So the function F (X, Y ) in the right h side satisfies the same hypergeometric differential equation for Z(X, Y ). Because F ( x, y ) x=y= = F (0, 0) =, we obtain the required equality. 4 Main theorem Theorem 4. We have in a neiborhood of (x, y) = (, ). M(, x, y) = H(x, y) = F (/, /, /, ; x, y ). 5
Set ϕ(x, y) = H(x, y)/f ( x, y ), Ψ(, x, y) = (α, β, γ). According to the functional equation, we have By the iteration of this argument we obtain ϕ(x, y) = ϕ(β/α, γ/α). ϕ(x, y) = ϕ(, ) =. 5 M(, x, y) as a period integral According to Appell we have integral representations for F (a, b, b, c; x, y) as follows: Theorem 5. (Appell [App]) () If we have R(a) > 0, R(c a) > 0, it holds F (a, b, b Γ(c), c; x, y) = u a ( u) c a ( xu) b ( yu) b du Γ(a)Γ(c a) 0 Γ(c) = u b+b c (u ) c a (u x) b (u y) b du, ( x <, y < ). Γ(a)Γ(c a) (2) If we have R(b) > 0, R(b ) > 0, R(c b b ) > 0, x <, y <, it holds F (a, b, b Γ(c), c; x, y) = Γ(b)Γ(b )Γ(c b b u b v b ( u v) c b b ( xu yv) a dudv. ) u,v, u v 0 So our M(, x, y) has expressions as period integrals for a family of Picard curves at the same time for a family of certain elliptic K surfaces (see [S]). Theorem 5.2 We have M(, x, y) = Γ( )Γ( 2 ) v = u(u )(u + x )(u + y ), w = uv( u v)( u( x ) v( y )) for x <, y <. Where, we choose the real positive branches of v w for real x, y. du v = dudv (Γ( )) u,v, u v 0 w, Remark 5. If we put x = y, our M councides with the cubic AGM discovered by Borweins [B-B]. References [App] P. Appell, Sur les Fonctions hypérgéometriques de plusieurs variables les polynomes d Hermite et autres fonctions sphériques dans l hyperspace, Gauthier-Villars, Paris, 925. [B-B] J.M. Borwein P.B.Borwein, A cubic counterpart of Jacobi s identity the AGM, Trans. Amer. Math. Soc., 2 (99), no. 2, 69 70. [B-M] J.B. Bost J.F. Mestre, Moyenne Arithmético-Géoétrique et Périodes des Courbes de Genre et 2, Gaz. Math. No. 8 (988), 6 64. [K-S] K. Koike H. Shiga, Isogeny formulas for the Picard modular forms a generalization of the cubic AGM, Tech. Rep. Chiba univ. 2005. 6
[R] F. Richelot, Essai sur une méthode générale pour déterminer la valeur des intégrales ultraelliptiques, fondée sur des transformations remarquables de ces transcendantes. C.R.Acad. Sci. Paris,2(86),622-627. [S] [Y] H. Shiga, On the representation of the Picard modular function by θ constants I-II, Pub. R.I.M.S. Kyoto Univ. 24(988), -60. M. Yoshida, Fuchsian differential equations, Aspects of Mathematics, Vieweg, Braunschweig(987). 7