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Cubic Theta Functions

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Cubic Theta Functions Cubic Theta Functions Daniel Schultz July 13, 2011 Abstract P m2+mn+n2 m n The theory of the cubic theta function q z1 z2 is developed analogously to the theory of the classical Jacobian theta functions. Many inversion formulas, addition formulas, and modular equations are derived, and these results extend previous work done on cubic theta functions. Introduction For any positive integer n, the rising factorial is defined by Γ(a + n) (a) := = a(a + 1) ··· (a + n − 1), n Γ(a) and the Gaussian hypergeometric function is defined by 1 X (a)n(b)n F (a; b; c; z) := zn. 2 1 (c) (1) n=0 n n In his paper [20] on modular equations and approximations to π, Ramanujan gives two series for π that \belong to the theory of q2" where 1 2 2! 2π 2F1 ; ; 1; 1 − k q (k2) = exp −p 3 3 . (0.1) 2 1 2 2 3 2F1 3 ; 3 ; 1; k Such an expression is an analogue of the classical relation between the nome q and modulus k2 of the elliptic curve y2 = x(1 − x)(1 − k2x) given by 1 1 2! 2F1 ; ; 1; 1 − k q = exp −π 2 2 (0.2) 1 1 2 2F1 2 ; 2 ; 1; k in which the hypergeometric function 1 1 K (z) := F ; ; 1; z (0.3) 2 2 1 2 2 has been replaced by 1 2 K (z) := F ; ; 1; z . (0.4) 3 2 1 3 3 1 Ramanujan gives several results in the theory of q2, and among the most interesting are several modular equations and an analogue of a inversion formula for the classical Jacobian elliptic functions. The latter (Theorem 8.1 in [3]) gives the solution for φ when p Z φ cos 1 sin−1 ( m sin φ) u = 3 dφ (0.5) p 2 0 1 − m sin φ by 1 dφ 3 X xn = , (0.6) du K (m) 1 + qn + q−n 3 n=−∞ where x = exp 2iu , and q = q (m) is given by (0.1). Here it seems more natural to use m K3(m) 2 2 to denote the modulus instead of using Ramanujan's k , and Ramanujan's \theory of q2" is more aptly called \the theory of cubic elliptic functions" or \the theory of signature 3". A proof of this inversion formula is given in [3] or [4], and a simpler derivation is given in [22]. Ramanujan defined a cubic modular equation of degree p as a relation between two moduli m1 and mp induced by the relation K (1 − m ) K (1 − m ) 3 p = p 3 1 , (0.7) K3(mp) K3(m1) and gave the following modular equations of degree 2, 5, and 11 respectively: x4 + y4 = 1, (0.8) x4 + 3x2y2 + y4 = 1, (0.9) p 2 4x2y2 + 3 3xy x2 + y2 + x2 + y2 = 1, (0.10) 1=12 1=12 1=12 1=12 where x = m1 mp and y = (1 − m1) (1 − mp) in all cases. Ramanujan's modular equations may be found in Chapter 33 of [3]. An example of a modular equation of degree 23 that will be proved in this paper is p 2 44x2y2 + 15 3xy x2 + y2 + 2 x2 + y2 = p p p q p p (0.11) 2−3 xy 3x2 + 4xy + 3y2 4 3x2 + 13xy + 4 3y2. Thus we see that the theory of the cubic elliptic functions is strikingly similar to the classical theory of elliptic functions; there are modular equations and elliptic functions that arise from the hypergeometric function K3(m) in place of the hypergeometric function K2(m). In fact, Ramanujan also developed results for the hypergeometric functions 1 3 K (z) := F ; ; 1; z (0.12) 4 2 1 4 4 and 1 5 K (z) := F ; ; 1; z , (0.13) 6 2 1 6 6 but the case of K3(z) is the one that gives the most interesting analogue to the classical theory of elliptic functions, and this is the case that is developed here. 2 Two central objects in the theory of elliptic functions (see, for example, [16] for a treatment of elliptic functions) are the elliptic integral 1 Z x dt u = p (0.14) πK2(m) 0 2 t(1 − t)(1 − mt) and the group of four theta functions 1=2 Θ (u; q) = Θ (u; q), 1 1=2 1=2 Θ (u; q) = Θ (u; q), 2 0 0 Θ (u; q) = Θ (u; q), 3 0 0 Θ (u; q) = Θ (u; q), 4 1=2 where 1 b X 2 Θ (u; q) = q(n+b) e2(n+b)πi(u+a). (0.15) a n=−∞ −1 The periods of the integral in (0.14) (without the factor (πK2(m)) ), are πK2(m) and iπK2(1 − m), hence the inversion of this integral gives rise to a doubly periodic function. More specifically, when u and x are related by (0.14) and q is given by (0.2) (and k2 = m), we have the fundamental inversion formulas which relate ratios of theta functions to x: Θ (0; q) Θ (u; q) p − 3 1 = x, Θ2(0; q) Θ4(u; q) Θ (0; q) Θ (u; q) p 4 2 = 1 − x, (0.16) Θ2(0; q) Θ4(u; q) Θ (0; q) Θ (u; q) p 4 3 = 1 − mx. Θ3(0; q) Θ4(u; q) These functions are the the Jacobian elliptic functions sn(ujm), cn(ujm), and dn(ujm) respec- tively. There is another important set of inversion formulas which relate m and K2(m) to the theta constants: 2 Θ1(0; q) = 0, 2 1=2 Θ2(0; q) = m K2(m), 2 (0.17) Θ3(0; q) = K2(m), 2 1=2 Θ4(0; q) = (1 − m) K2(m). With the last set of inversion formulas, the relation K (1 − m ) K (1 − m ) 2 p = p 2 1 , (0.18) K2(mp) K2(m1) which defines a modular equation of degree p in the classical theory, becomes (where q is 3 defined by (0.2)) 4 Θ1(0; q) m1 = 4 , Θ1(0; q) p 4 Θ1(0; q ) mp = p 4 , Θ1(0; q ) so that a modular equation is equivalent to an identity involving theta constants at q and qp. This modular equation is actually only part of the general pth order transformation for elliptic integrals, which asks for the relationship between x1 and xp induced by the equation Z x1 Z xp pK2(mp) dt dt p = p . (0.19) K2(m1) 0 2 t(1 − t)(1 − m1t) 0 2 t(1 − t)(1 − mpt) The quantity M = K2(m1) is known as the multiplier of degree p in the classical theory, and p K2(mp) the case p = 2 of this relationship is known as the Gauss or Landen transformation. In terms of a parameter t, we have t(t + 2) m = , 1 (t + 1)2 t2 m = , 2 (t + 2)2 (0.20) 2 t + 2 = , M t + 1 2 p p p 2 x1 1 − x1 x2 = p , M2 1 − m1x1 and this is a complete statement of the transformation of degree 2. The theta constants responsible for inverting K3(m) analogously to (0.17) were first identi- fied by the Borweins in [6] and Hirschhorn, Garvan, and Borwein in [15], where they introduced the notations (set ζ = exp 2πi=3 through out this paper) 1 X 2 2 a(q) = qm +mn+n , m;n=−∞ 1 X 2 2 b(q) = ζm−nqm +mn+n , m;n=−∞ 1 2 2 X m+ 1 + m+ 1 n+ 1 + n+ 1 c(q) = q( 3 ) ( 3 )( 3 ) ( 3 ) . m;n=−∞ They proved that these functions satisfy the identity a(q)3 = b(q)3 + c(q)3 (0.21) and the inversion formula c(q)3 K = a(q), (0.22) 3 a(q)3 hence these functions should be called \cubic theta functions". In order to produce satisfactory cubic analogues of the Landen transformation for elliptic integrals and differential equations and addition theorems satisfied by the Jacobian elliptic 4 functions, it seems that a group of nine theta functions in two analytic variables and the complex parameter q is necessary. As an example, we will define cubic theta functions of two 2πiu complex variables u = (u1; u2), four of which are (set zi = e i ) 1 X 1 2 1 1 1 2 m+ 1 n+ 1 1+m−n (m+ 3 ) +(m+ 3 )(n+ 3 )+(n+ 3 ) 3 3 Θ1(u; q) := ζ q z1 z2 , m;n=−∞ 1 X 1 2 1 1 1 2 m+ 1 n+ 1 (m+ 3 ) +(m+ 3 )(n+ 3 )+(n+ 3 ) 3 3 Θ2(u; q) := q z1 z2 , m;n=−∞ 1 X m2+mn+n2 m n Θ3(u; q) := q z1 z2 , m;n=−∞ 1 X m−n m2+mn+n2 m n Θ4(u; q) := ζ q z1 z2 , m;n=−∞ that satisfy 3 3 3 3 Θ1(u; q) + Θ3(u; q) = Θ2(u; q) + Θ4(u; q) , which gives a two variable generalization of (0.21), and is an analogue of the the equations sn(u)2 + cn(u)2 = 1, k2sn(u)2 + dn(u)2 = 1. The other five cubic theta functions are denoted by Θ11,Θ12,Θ14,Θ22,Θ44 and are defined in Section 2.3. We also have a triplication formula, b(q)c(q)2 Θ (3u; q3) = Θ (u; q)Θ (u; q)Θ (u; q)+ b(q3) 4 1 12 44 Θ2(u; q)Θ4(u; q)Θ22(u; q) + Θ3(u; q)Θ11(u; q)Θ14(u; q), that is analogous to the classical theta function identity 2 2 Θ4 0; q Θ4 2u; q = Θ3(u; q)Θ4(u; q), which is a key component in deriving the Landen transformation.
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