M ath. Res. Lett. 16 (2009), no. 3, c International Press THE APÉRY NUMBERS, THE ALMKVIST-ZUDILIN NUMBERS AND NEW SERIES FOR 1/π

Transcription

1 M ath. Res. Lett , no. 3, c International Press 009 THE APÉRY NUMBERS, THE ALMKVIST-ZUDILIN NUMBERS AND NEW SERIES FOR 1/π Heng Huat Chan and Helena Verrill Abstract. This paper concerns series for 1/π, such as Sato s series 1.4, and the series of H.H. Chan, S.H. Chan and Z.-G. Liu 1.8 below. The examples of Sato, Chan, Chan and Liu are related to two index subgroups of Γ These examples motivate us to look at a third subgroup of Γ We give a new method of constructing such series using the theory of modular forms and conclude our work with several new examples. For n 1, define 1.1 α n = 1. Introduction n n n + j. j j j=0 These numbers are known as Apéry s numbers, because of their appearance in Apéry s proof of the irrationality of ζ3 []. The first few terms are 1, 5, 73, 1445 this is sequence A00559 in [18]. Let q = e πiτ and ητ = q 1/4 n=1 1 q n be the Dedekind eta function. From [14, Theorem 5], it follows that if 1 η6τητ 1. t 1 τ = and F 1 τ = ητη3τ7 ητη3τ ητη6τ 5, then provided t 1 τ is sufficiently small, 1.3 F 1 τ = α n t n 1 τ. T. Sato [16] used 1.3 to obtain the following series for 1/π: π = α n n 1 1n 5. Sato s work motivated H.H. Chan, S.H. Chan and Z.-G. Liu [7] to consider 6 ητη6τ 1.5 t τ = and F τ = ητη3τ4 ητη3τ ητη6τ. Received by the editors June 16,

2 406 HENG HUAT CHAN AND HELENA VERRILL It turns out that when t τ is sufficiently small, we have the following analogue of 1.3: 1.6 F τ = β n t n τ where β n = 1 n β n, where n n 1.7 βn = j p+q+r+s=n j=0 j j n j = n j p+q+r+s=n where n = a 1, a,, a r Note that the second equality here follows from writing n = n p, q, r, s k, j n, p, q, r, s n! a 1!a! a r!, with a 1 + a + a r = n. k+j=n p+q=k k p, q r+s=j j r, s and using the identity s s i=1 i = s s. The first few values of βn are 1, 4, 8, 56; this is sequence A00895 in [18]. An example given in [7] of a series for 1/π in terms of the β n is 8 n = β n 1 + 5n. 3π 64 The functions t 1 τ and t τ are Hauptmoduls of Γ and Γ respectively, which are two of the three index subgroups lying between Γ 0 6 and its normalizer in SL R. These are given by Γ 0 6 +k = Γ 0 6, w k, where k =, 3, 6, and w = 1 1 6, w3 = , w6 = It is natural to ask whether there exist analogues of 1.3 and 1.6 for the third such group, Γ We will show that the answer is affirmative. In fact, when 4 η3τη6τ t 3 τ =, and F 3 τ = ητητ3 ητητ η3τη6τ, and t 3 τ sufficiently small, we show that F 3 τ = γ n t n 3 τ, where γ n are the Almkvist-Zudilin numbers [1] given by n 1.9 γ n = 1 n j 3n 3j 3j! n j! 3 3j j=0 n + j j The first few terms of this sequence are 1, 3, 9, 3, 79, 997. In this paper, we prove that we have the following analogues of 1.4 and 1.8: π = n 1 γ n 4n + 1, 81.

3 π NEW SERIES FOR 1/π 407 = γ n 4n + 1 n 1. 7 Experimentally, we find further series such as = γ n n π n 6. 9 In this article, in the course of proving 1.10 and 1.11, we give a general strategy for finding such series Theorem.1. A number of other series are derived experimentally, and they are given in Tables 1, and Comparison with earlier works. Our methods are similar to those of [7]. Our series also have the form π = a n An + Bt n 0, where A, B are constants, and t 0 is the value of a Hauptmodul t for some genus zero subgroup of SL R at some point z 0 in the upperhalf complex plane. Our t plays the same role as X in [7,.7], and e π N/s of [7,.1] is now replaced by expπiz 0, i.e., setting z 0 of this paper to z 0 = i N/s + 1/, for integers N and s, will lead towards the formulas of [7,.1]. However, in this paper, z 0 does not have to have this form. In theory, z 0 just has to be in an imaginary quadratic extension of Q, though in practice, to actually be able to compute the values of tz 0, the possible values of z 0 are still restricted. The series 1.10, proved in Theorem 3.14 corresponds to z 0 = i/, and 1.11, proved in Theorem 3.15 corresponds to z 0 = i + 1/ see Lemma 3.1. In the experimental results in Tables 1, and 3, we take z 0 = i l/6, for various integers l. Another difference of this work compared with [7] is that in the earlier work, the constants A and B in 1.13 were given in terms of the derivative of a modular form fτ with respect to the variable q, where q = expπiτ. In this paper, the model for our formulas is.17, where the derivative is now with respect to the variable t. In this paper, the ingredients of the main result, Theorem.1, are a Hauptmodul t, a modular form of weight, F, and an order element g of SL R. In order to apply the theorem to obtain explicit formulas for 1/π, we need to be able to determine the value of tz 0, where z 0 is the fixed point of g, and we need to be able to express the weight 0 modular function R = F,g /tf as an algebraic function of t, where F,g is given by the usual action of SL R on modular forms.1. This determines the value of the term p g in 3.1, 4., 5.1. We also need to express Θt/F as an algebraic function of t, where Θ = q d dq. Thus, for fixed t and F, once t 0 and R are determined, in order to determine a series for 1/π corresponding to a choice of g reduces to determining the value of tz 0 and F,g /F as an algebraic function of t, e.g., as in Lemma 3.9. An empirical process to find p g can be obtained from [4]. In the examples in Tables 1, and 3, one can observe that in most cases t 0 is a unit. This phenomenon is explained in [8, Cor. 5., page 87].. A strategy for finding series for 1/π using the theory of modular forms.1. Modular forms.

4 408 HENG HUAT CHAN AND HELENA VERRILL Let Γ be a subgroup of SL R, commensurable with SL Z, and χ a character of Γ of finite order, namely, a group homomorphism χ : Γ C with finite image. A modular form of weight k, character χ for Γ SL R is a holomorphic function F z on the upper half complex plane h satisfying the following conditions: a For every g = a b c d GL + R, where F k,g z = χgf z,.1 F k,g z = detg k/ jg, z k F gz, with jg, z = cz + d and gz := az + b cz + d. For the rest of this article we always assume detg = 1. b F z is holomorphic at cusps. The space of all such modular forms is denoted by M k Γ, χ, and χ is omitted when it is the trivial character. We write A k Γ, χ to denote the space of meromorphic forms satisfying F k,g z = χgf z. See for example [1,.1] or [17, Chapter ] for complete definitions... Differentiation and the appearance of π. In general, the derivative of a modular form is not a modular form []. Differentiating.1 with respect to z, and assuming detg = 1, gives d dz F k,gz = 1 d jg, z k+ dz F gz ck F gz jg, z k+1 i.e., writing c = c g, f z = df dz, and dividing through by F k,g, we have F k,g = F z z k+,g c gk. jg, z F k,g. This shows that if F 0,g = F, then F z = F z,g, i.e., the derivative of a weight k = 0 function has weight. We will be interested in the case k =. Provided that Γ, and χ = 1, then a modular form F z M kγ, χ can be written as a function of qz = expπiz, and we define F q to be a function such that F z = F qz. The condition that F be holomorphic at cusps means that F q has a Taylor series expansion F q = n 0 a n q n for some constants a n. This function is called the q-expansion of F. When q d dq is applied to a Taylor series in q with integral coefficients, the resulting series also has integral coefficients. This is our motivation for setting Θ = 1 d πi dz,

5 NEW SERIES FOR 1/π 409 so that ΘF z = qz d dq F qz. In terms of Θ,. becomes.3 Θ F k,g = ΘF k+,g This equation will lead to a series for 1/π. c g k jg, zπi F k,g..3. Genus zero subgroups of SL R commensurable with SL Z. We denote by h the upper-half complex plane, and h = h Q { }. If the Riemann surface XΓ := Γ \ h has genus zero then there is a function t, known as a Hauptmodul, satisfying A 0 Γ = Ct. In other words, any element of A 0 Γ can be written as a rational function in t. In this case we say that Γ has genus 0. For the remainder of this article we will always assume Γ has genus zero and that t is a Hauptmodul associated with Γ..4. Power series for F t. If Γ and χ = 1, so that F M kγ, χ and t A 0 Γ, χ both have q-expansions. If we also have ti = 0, then by a series inversion, for Imz sufficiently large, we have F z = F tz, where for some sequence γ n,.4 F t = γ n t n. The γ n satisfy a recurrence relation, which follows from the existence of differential equations solved in terms of modular forms [19], [1]..5. Conjugation of subgroups of SL R. Let g, h SL R. Since the action defined by.1 is a group action, we have If F k,h = χhf, then F k,g k,g 1 hg = F k,h k,g. F k,g k,hg = χhf k,g, where h g = g 1 hg. This shows that if F M k Γ, χ, then where Hence F k,g M k g 1 Γg, χ g 1, χ g 1 h = χghg 1..5 F k,g F where Γ g = g 1 Γg. We define F g by A 0 Γ Γ g, χ g 1 χ 1,.6 F g := F k,g F. When Γ has genus 0, Hauptmodul t, if Γ g = Γ and χ g 1 = χ, then F g is a rational function of t, since in this case.5 says that F g A 0 Γ. Provided that Γ and Γ g are commensurable, F g z will be an algebraic function of t, since F g A 0 H where

6 410 HENG HUAT CHAN AND HELENA VERRILL H Γ Γ g is the kernel of χ g 1 χ 1, which is a subgroup of Γ of finite index. We will suppose that we are in a situation where there is some algebraic function P g with.7 P g tz = F g z..8 Applying Θ ln to.6 gives From.3 we have.9 Θ F k,g ΘF g F g = ΘF k,g F k,g F k,g = ΘF k+,g F k,g ΘF F. Together with.8, this gives the following identity c g k ΘF ΘF.10 πijg, z = F F,g c g k πijg, z. + ΘF g F g. Note that we have used the fact that A k 1,g B = A k,g B k 1 k,g. Take F t as in.4. In a neighbourhood of z = i where F z = F tz we have.11 From.7 we have.1 ΘF z F z ΘF g z F g z = Θtz F z = Θtz substituting.1 and.11 into.10 gives.13 c g k Θtz πijg, z = d F z F dt tz Θtz F z,g d dt F tz. d dt P g tz P g tz d dt F tz + d dt P g tz P g tz F z..6. The case k =. From., if t A 0 Γ then Θt A Γ, so if F M Γ, χ, then A 0 Γ, χ. Suppose that there is an algebraic function R with.14 Θtz tzf z = Rtz. Substituting.14 and.4 into.13, we obtain Rtz nγ n tz,g n Rtz n + tz d dt P g tz P g γ n tz n tz.15 = c g πijg, z. Now let z 0 be a fixed point of g and set t 0 = tz 0. Then.15 becomes c g [ ].16 πijg, z 0 = Rt 1 0 n 1 jg, z 0 + t 0 d dt P g t 0 P g γ n t n 0. t 0 Θtz tzf z

7 NEW SERIES FOR 1/π 411 In the case that g = I, we have jg, z 0 = i and so we obtain c g.17 πrt 0 = n + t 0 d dt P g t 0 P g γ n t n 0. t 0 To summarize, we have now proved the following: Theorem.1. Let tz be a Hauptmodul for some genus 0 subgroup Γ SL R commensurable with SL Z, and let F M Γ, where Γ is a finite index subgroup of Γ. Let g SL R with g = I. Let z 0 be a point in the upperhalf complex plane fixed by g. Let t 0 = tz 0. Suppose that ti = 0. Let F t = n 0 γ nt n be a function such that for some M > 0, provided Imz > M, we have F tz = F z. Suppose that there are algebraic functions Rt and P g t so that Rtz = and P g tz = F z,g F z. Then provided Imz 0 > M,.17 holds. Θtz tzf z 3. Series for 1/π arising from modular forms for Γ We now take Γ = Γ 0 6 +, the group generated by Γ 0 6 and w, where Γ 0 6 = { a b c d SL Z c 0 mod 6 }, w = l/6 We will apply Theorem.1 with g =, F = F 3, t = t 3, where 6/l t 3 τ = 4 η6τη3τ = q + 4q + 18q 3 + 5q q 5 + ητητ F 3 τ = ητητ3 η6τη3τ = 1 3q 3q + 15q 3 3q 4 18q 5 + Lemma 3.1. F 3 M Γ 0 6 +, χ, where χg = 1 if g Γ \ Γ 0 6, and χg = 1 otherwise, and t 3 is a Hauptmodul for Γ Proof. Since ητ is a modular form of weight 1/ for SL Z, ηnτ is a modular form of weight 1/ for Γ 0 N, and so F 3 is a modular form of weight for Γ 0 6. Its transformation behavior under the action of w can be determined by using Dedekind s functional equation [3, Theorem 3.4] az + b 3. η = ηz icz + d ɛ sa,b,c,d, cz + d where ɛ = expπi/4, sa, b, c, d = a + d c c 1 r dr + 1 c c r=1 [ dr c ] 1 and a, b, c, d are integers with ad bc = 1, and c > 0. The transformation properties of t 3 can similarly be checked; t 3 is also given in Conway and Norton s tables [9] as a Hauptmodul for Γ

8 41 HENG HUAT CHAN AND HELENA VERRILL Lemma 3.. If δ n is a sequence such that in a neighborhood of τ = i, F 3 τ = δ n t n 3 τ, then δ n satisfy a recurrence relation 3.3 n 3 δ n + n 17n 7n + 3δ n n 1 3 δ n = 0. Proof. First note that F 3 t 3 = s r, where s and r are modular functions for Γ 0 6 defined by 3.4 sτ = η3τ6 ητ ητ η6τ 3, rτ = ητ3 η6τ 9 ητ 3 η3τ 9. Let Sr be a series defined by Sr = ζ n r n where ζ n = n j=0 3 n. j Then by interpreting s as a differential form on a family of elliptic curves, as in e.g., [14] near τ = i we have sτ = Srτ see [0, Table ]. The sequence ζ n is classical, see e.g., Sloane s sequence A00017 [18] for several references and Sr satisfies the differential equation 3.5 8r 1r + 1Θ rs + r16r + 7Θ r S + r4r + 1S = 0, where Θ r = r d dr. We will now substitute r for r and s for S. By replacing s by r 1/ u, we find that u = r 1/ s satisfies Eu = 0 where 3.6 E = 8r 1r + 1Θ r + 8r + 1Θ r r + r 1. Note that u = t 3 F 3. Since r is a Hauptmodul for Γ 0 6 taken from [9], our Hauptmodul t = t 3 for Γ is given by a degree rational function in r, namely r 3.7 t = r r. Make a substitution to write 3.6 in terms of t, which transforms E to [ Θ t + r + 18r ] 18r 1 8r + 1 Θ t r + 18r 18r + r 1 48r + 1, which can be written as 3.8 Θ t + [ ] 7t t Θ t + 14t + 1 6t t + 14t + 1. A substitution shows that K = t 1/ u = F 3 satisfies GK = 0 where 3.9 G := Θ t + t14θ t + 7Θ t t Θ t + 1. As explained in [1, Proposition 9], the symmetric square of G is 3.10 D = Θ 3 t tθ t + 17Θ + 7Θ t Θ + 1 3, and so K = δ nt n satisfies DK = 0. By the Frobenius method, this is equivalent to the δ n satisfying the recurrence relation 3.3 [10, Chapter XVI, 16.11].

9 NEW SERIES FOR 1/π 413 Lemma 3.3. With δ n as in Lemma 3., for n 0 we have δ n Almkvist-Zudilin numbers γ n are given by 1.9. = γ n, where the Proof. By Lemma 3., the δ n satisfy the recurrence 3.3. According to [1, δ p. 498], the n 0 γ nt n satisfy the differential equation 3.10, which is equivalent to the γ n satisfying 3.3. In more detail, the summand of γ n has the simple form j c n,j = 3 n n + j! 1 j! 4, n 3j! 7 and so Zeilberger s telescoping sum algorithm for finding recurrences, described in [15, Chapter 6], applies, and the book s accompanying Maple package Ekhad, computes the easily verifiable relation 81 n c n,j + n + 3 7n + 1n + 17 c n+1,j + n + 3 c n+,j = Gn, j + 1 Gn, j, 4n + 5 j 4 where Gn, j = 34 n + 3j n + 1 3j c n,j, which via Zeilberger s method i.e., summing over all j leads to after substituting n for n the recurrence 3.3. A substitution of the inversion of the q-expansion of t 3 into the q-expansion for F 3 shows that the first few δ n are 1, 3, 9, 3, 79, and so are equal to the γ n, and so all terms agree. Lemma 3.4. With the Almkvist-Zudilin numbers γ n radius of convergence 1 9. Proof. This follows from [10, Chapter XVI, 16.] by writing 3.10 as as in 1.9, γ nt n has 81t + 14t + 1t 3 d3 dt t + 1t + 1t d dt + 1t + 17t + 1t d + 37t + 1t dt and noting that the roots of 81t + 14t + 1 = 0 have absolute value 1 9. Lemma 3.5. As multivalued functions on XΓ 0 6, F 3 and t 3 have orders of vanishing at cusps of Γ 0 6, denoted by O c, as in the following table: cusp c 0 1/ 1/3 width O c t O c F Proof. Quoting [11, 5], the Fourier series of η g z at the cusp a c is η g z 1 a b = C exp πiz s gcdt j, c r j G a/c z, c d 4 t j where C is a constant and G a/c z is holomorphic and nonvanishing at i. This formula allows us to compute the leading term of the q = expπiz expansion. If this is q d for cusp c, then we must multiply d by the cusp width to obtain the order of vanishing. j=1

10 414 HENG HUAT CHAN AND HELENA VERRILL Lemma 3.6. Let F and t be defined as in 3.1. Using the notation of.14, 3.11 Rt = 81t + 14t + 1 Proof. Since F z M Γ 0 6, tz A 0 Γ 0 6 and Θtz A Γ 0 6, we have Rtz A 0 Γ 0 6, and so Rtz may be expressed as a rational function in terms of the Hauptmodul r given by 3.4 for Γ 0 6. Since Θtz has exactly the same orders of vanishing at cusps as t, the orders of vanishing of Rtz at cusps are determined by those of F, so from Lemma 3.5, Rtz has poles of order one at 0 and 1, and no other poles since t is holomorphic on h, any poles of Rtz are at cusps. Thus R is a degree rational function of r. By comparing q expansions, we find that R = 1+8r 1+r1 8r. Using expression 3.7 leads to Corollary 3.7. Let l be a positive integer. Let the Almkvist-Zudilin numbers γ n be as in 1.9, and t 0 = ti l/6, where t is as in 3.1, and let p g = t0 d dt P g t 0 P g t 0 with P g as in.7. Then provided t 0 < 1/9, π 6 l81t t = n + pg γn t n 0 Proof. This is an application of Theorem.1 with F and t as in 3.1 and g = 0 l/6. Values of γ n come from Lemma 3.3, Rt 0 is given by Lemma 3.6, 6/l 0 and the radius of convergence is in Lemma 3.4. Example 3.8. Since t 3 is an eta product, the value of t 0 in 3.1 can be found to many decimal places using a computer program such as PARI [13]. Now 3.1 has the form At 0 = Bt 0 + p g Ct 0, so an approximation to t 0 allows us to determine the expected value of p g. Experimentally, we can search for values of l where t 0 and p g both appear to be roots of quadratic equations, obtaining the series in Table 1. We will only prove the case l = 3.

11 NEW SERIES FOR 1/π 415 l = l = 3 l = 5 l = 7 l = 13 l = π π π π = n + 1 γ n 10π π = n + 1 = n γ n 0 = n γ n 7 = n γ n 65 = n γ n 17 n γ n n n n n n Table 1. Series for 1 π in terms of the Almkvist-Zudilin numbers Lemma 3.9. With t and F as in 3.1, g = 0 1, and P = F3,gτ 0 F 3τ, p g t := t d dt P t P t P 3 = 7t1 P + P, = P P + 1 P P + 3. Proof. We have gz = 1 z, and by Dedekind s functional equation, with a numerical computation to be sure of the sign of the square root, P tτ = F 3,g τ F 3 τ = 3 η6τη3τ η τ 3 η τ 3. So, P 3τ = 3 xτ, where xτ = ητητ η18τη9τ. Note that xτ4 t3τtτ = 1. Since x is a Hauptmodul for Γ 0 18 [9], t for Γ 0 6, and [Γ 0 6 : Γ 0 18] = 3, it is possible to write t as a degree 3 rational function in x, which we can determine, by comparision of q-expansions, to be tx 3 = x + 3x + 9. From this we have 1 = xτ 4 t3τtτ = xτt3τxτ + 3xτ + 9. By substituting xτ = 3 P 3τ, and replacing τ by τ/3, this gives the first statement, from which the second follows. Lemma With t = t 3 we have t 1 6τ tτ 81 = 1. Proof. This follows from Dedekind s functional equation for η. i To determine t, we use the relationship between Γ and Γ 0 +. Lemma Γ 0 + has a Hauptmodul t A satisfying 3.13 t A = where t is as in t4, t

12 write t as a degree 3 rational function in x, which we can determine, by comparision of q-expansions, to be tx 3 = x + 3x + 9. From this we have 1 = xτ 4 t3τtτ = xτt3τxτ +3xτ+9. By substituting xτ = 3 P3τ, and replacing τ by τ/3, this gives the first statement, from which the second follows. 416 HENG HUAT CHAN AND HELENA VERRILL Lemma With t = t 3 we have t 1 6τ tτ 81 = 1. Proof. From [9] we have t B = t A /t A, where t B = ητ/ητ 4. From this Proof. we find This enough follows terms from of the Dedekind s q-expansion functional to determine equation the for relation η. between t and t A, which has degree 4 in t, since [Γ 0 + : Γ ] = 4. i To determine t, we use the relationship between Γ and Γ 0 +. m 0 1 1=, m 0 = 1 0 1, m 3= i G m 1G G i 1 0 = m G m 3G 1 3 +i i Figure Figure 1. Fundamental 1. Fundamental domain domain G for G for Γ Γ 0 + and domain for 0 + and domain for Γ Γ in terms of G. Matrices I, 1, m 3 are coset representatives in terms of G. Matrices I, m 1, m, m 3 are coset representatives for Γ for Γ in Γ 0 +. Matrices written near elliptic points in Γ 0 +. Matrices written near elliptic points fix these fix these points, points, and and identify identify adjacent adjacent edges. edges. The The edge edge identifying identifying matrices matrices in each in each case case generate generate the the groups. groups. Matrices Matrices are are only only given given up to up a to real a real constant. constant. 0 1 Lemma 3.1. With t as in 3.1, we have 1 + i t = 1 i 7, t = 1 81 and i t = 1 18 Proof. Motivated by 3.13, define f α y := 1 + 7y 4 yα. Note that i and i are elliptic points of Γ 0 +, but not of the subgroup Γ This is illustrated in Figure 1 showing fundamental domains for these groups, from which we see that the image of 1 + i/ in XΓ 0 + has only one preimage in XΓ 0 6 +, whereas the image of i/ has three preimages, two being elliptic points, the other mapping to i/ with multiplicity. This means that f α y = 0 has a quadruple root when α = t A 1+i/, and a double root when α = t A i/. The discriminant of f α y is 3 7 α 3 α 8, and so for f α y = 0 to have a multiple root, we require α = 0 or α = 8. If α = 0, we have a quadruple root, y = 1/7, so t A 1 + i/ = 1/7. With α = 8, we have f 8y = 81y 1 81y + 14y + 1,

13 NEW SERIES FOR 1/π 417 from which we obtain ti/ = 1/81, as well as the fact that the values of t at the elliptic points of Γ are roots of 81t +14t+1. The value for ti/ 18 = ti /6 now follows from Lemma Lemma With P as in Lemma 3.9, P i/ = 1. Proof. From Lemma 3.9, we have P 3 = 7t1 P + P, and P 3τ 4 = 3 4 t3τtτ. For τ = i/ 18, the latter implies P i/ 4 = 1, by Lemma The first relation, together with ti/ = 1/81 Lemma 3.1 implies P + 13P i P + 1 = 0, so the only possibility is P = 1. Theorem With the Almkvist-Zudilin numbers γ n as in 1.9, we have 9 n 1 3π = γ n 4n Proof. We apply Corollary 3.7 with l = 3. From Lemma 3.1, t 0 = 1/81, and from Lemma 3.13 and Lemma 3.9, p g = 1/. Plugging these values into 3.1 and multiplying by, gives the result. In Lemma 3.1, we saw that t1+i/ = 1/7. Since m = fixes 1+i/, we apply Theorem.1 with g = m. The function Rt is still as in Lemma 3.6, and R 1 7 = 4 3, and in Lemma 3.16 below we show that p 3 g = p m = 1/. Plugging these values into.17 we obtain the following Theorem π = γn 4n n. Lemma With t and F as in 3.1, m = 1 1 F 1, and P = 3,m τ F 3τ, p m t := t d dt P t P t P 3 α 3 = 81tα P + 3αP + 9 where α = 3 i. Moreover, p m t1 + i/ = 1. = α P + 3αP + 9 α 3 P + 6αP + 7, Proof. First note that mτ = g 1 τ, mτ = g u, 6mτ = g 3 u, where g 1 = 1 1 1, g = 3 1 and g3 = are all in SL Z, and u = 1 3 z + 1. From Dedekind s functional equation, 3., it follows that F 3,m z = 3 i ηzηz3 ηuηu. Since z = 3u 1, we have P m z = η9uη18u ηuηu = 3 i xu, with x as in Lemma 3.9. Proceeding as in Lemma 3.9, using t3u + 1 = t3u = xu 4 tu 1, we obtain the expression relating P and t, from which the second expression follows.

14 418 HENG HUAT CHAN AND HELENA VERRILL By Lemma 3.1, t1 + i/ = 1 7, and y = αp 1/7 satisfies y3 + 3y + 3y + 9 = y + 3y + 9 = 0. One can check numerically that the correct root is αp 1/7 = 3, from which we obtain p m 1/7 = 1/. 4. Series for 1/π arising from modular forms for Γ Exactly the same method as used for Γ and F 3, t 3 in Section 3 can be applied to Γ and F, t given by 1.5, using the same g. The value of r, given by Lemma 3.6 for the t 3, F 3 case, is now replaced by 4.1 Rt = 4t + 116t + 1, 4. 6 π l4t t = n + p g β n t n 0, As explained in Example 3.8, one can now search for series for 1/π. Experimentally, we obtain the series in Table. The exact details of proof should be similar to the above methods, with the l = case being the simplest. 4 l = 3π = n + n 1 β n 3 3 l = 3 l = 4 l = 5 l = 7 l = 8 l = 10 l = 13 l = = n + 1 β n 5π = n + 1 β n 3π = n + 1 β n 5 8 = n β n = n β n = n β n = n β n 1 8 = n β n π π π π π π n n n n n n n n 5. Series for 1/π arising from modular forms for Γ For F 1 and t 1 as in 1., we apply Theorem.1 with 0 l/6 g = 6/l 0

15 NEW SERIES FOR 1/π 419 Now R is given by Rt = t 34t + 1, and we have n π l t 0 34t = + p g α n t n 0, where t 0 = t 1 i/ 6l, and α n are the Apéry numbers, defined by 1.1. Experimentally, we obtain series as in Table 3. For l = 5 we recover Sato s example 1.4 cf. [16]. l = l = 3 l = 5 l = 7 l = 13 l = = n α n 3 3n 6π = n + 1 8π α n 3 4n π = n α n 5 4n = n α n n 7π = n α n 6 5 4n 13π = n α n 1 1n 816π 68 Acknowledgements We thank W. Zudilin for pointing out some errors in Table 3 of an earlier version of this paper. The first author is supported by National University of Singapore Academic Research Fund grant number R The second author author is partially supported by grants LEQSF RD-A-16 and NSF award DMS References [1] G. Almkvist and W. Zudilin. Differential equations, mirror maps and zeta values. In Mirror Symmetry V, N. Yui, S.-T. Yau, and J.D. Lewis eds., AMS/IP Studies in Advanced Mathematics , International Press and Amer. Math. Soc., [] R. Apéry. Interpolation de fractions continues et irrationalité de certaines constantes, In Mathematics, CTHS: Bull. Sec. Sci., III, pages Bib. Nat., Paris, [3] T.M. Apostol. Modular functions and Dirichlet series in number theory, volume 41 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, [4] B.C. Berndt and H. H. Chan Ramanujan and the Modular j-invariant, Canadian Mathematical Bulletin, , [5] J.M. Borwein and P.B. Borwein. Ramanujan s rational and algebraic series for 1/π, J. Indian Math. Soc. N.S , [6] B.C. Berndt. Ramanujan s notebooks. Part IV, Springer-Verlag, New York, [7] H.H. Chan, S.H. Chan, and Z.-G. Liu. Domb s numbers and Ramanujan-Sato type series for 1/π, Adv. Math ,

16 40 HENG HUAT CHAN AND HELENA VERRILL [8] H.H. Chan and S.S. Huang, On Ramanujan-Gollnitz-Gordon continued fraction, The Ramanujan J , [9] J.H. Conway and S.P. Norton. Monstrous moonshine, Bull. London Math. Soc , [10] E. L. Ince. Ordinary Differential Equations. Dover Publications, New York, [11] Y. Martin. Multiplicative η-quotients, Trans. Amer. Math. Soc , [1] T. Miyake. Modular forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 006. Translated from the 1976 Japanese original by Yoshitaka Maeda. [13] PARI Group, Bordeaux. PARI/GP, version.3.0, 006. available from u-bordeaux.fr/. [14] C. Peters and J. Stienstra. A pencil of K3-surfaces related to Apéry s recurrence for ζ3 and Fermi surfaces for potential zero, Arithmetic of complex manifolds Erlangen, 1988, volume 1399 of Lecture Notes in Math. pages Springer, Berlin, [15] M. Petkovšek, H. S. Wilf, and D. Zeilberger. A = B. A K Peters Ltd., Wellesley, MA, With a foreword by Donald E. Knuth, With a separately available computer disk. [16] T. Sato. Apéry numbers and Ramanujan s series for 1/π, Abstract of a talk presented at the annual meeting of the Mahtematical Society of Japan, 8-31 March 00. [17] G. Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, Reprint of the 1971 original, Kanô Memorial Lectures, 1. [18] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences, 006, Published online at njas/sequences/. [19] P. Stiller. Special values of Dirichlet series, monodromy, and the periods of automorphic forms, Mem. Amer. Math. Soc , iv+116. [0] H. Verrill. Some congruences related to modular forms, MPIM Preprint Series, MPIM1999-6, available from [1] Y.F. Yang. On differential equations satisfied by modular forms, Math. Z , [] D. Zagier. Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci , K. G. Ramanathan memorial issue. Department of Mathematics, National University of Singapore, Science Drive, Singapore address: matchh@nus.edu.sg Department of Mathematics, Louisiana State University, Baton Rouge 70803, Louisiana address: verrill@math.lsu.edu