A three terms Arithmetic-Geometric mean

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1 A three terms Arithmetic-Geometric mean Kenji Koike Faculty of Education University of Yamanashi Takeda 4-4-7,Kofu Yamanashi , Japan Hironori Shiga Inst. of Math. Physics Chiba University Yayoi cho, Inage ku Chiba , Japan june 04, 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.

2 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

3 Proposition. We have b c < 0.982(a c), a b < (a c), a c < (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 < (a c). a c < (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 ).

4 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

5 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

6 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, [B-M] J.B. Bost J.F. Mestre, Moyenne Arithmético-Géoétrique et Périodes des Courbes de Genre et 2, Gaz. Math. No. 8 (988), [K-S] K. Koike H. Shiga, Isogeny formulas for the Picard modular forms a generalization of the cubic AGM, Tech. Rep. Chiba univ

7 [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), [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