(6n + 1)( 1 2 )3 n (n!) 3 4 n, He then remarks that There are corresponding theories in which q is replaced by one or other of the functions

Transcription

1 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES Bruce C Berndt, S Bhargava, Frank G Garvan Contents Introduction Ramanujan s Cubic Transformation, the Borweins Cubic Theta Function Identity, the Inversion Formula The Principles of Triplication Trimidiation The Eisenstein Series L, M, N 5 A Hypergeometric Transformation Associated Transfer Principle 6 More Higher Order Transformations for Hypergeometric Series 7 Modular Equations in the Theory of Signature 8 The Inversion of an Analogue of Kk in Signature 9 The Theory for Signature 0 Modular Equations in the Theory of Signature The Theory for Signature 6 An Identity from the First Notebook Further Hypergeometric Transformations Concluding Remarks Introduction In his famous paper [Ramanujan], [RamanujanCP, pp 9], Ramanujan offers several beautiful series representations for /π He first states three formulas, one of which is π n0 where a 0, for each positive integer n, 6n + n n! n, a n aa + a + a + n He then remarks that There are corresponding theories in which q is replaced by one or other of the functions q exp π K /K, q exp πk /K, q exp πk /K, where K F, ; ; k, K F, ; ; k, Typeset by AMS-TEX

2 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN K F 6, 5 6 ; ; k Here K j K jk, where j, k k, 0 < k < ; k is called the modulus In the classical theory, the hypergeometric functions above are replaced by F, ; ; k Ramanujan then offers 6 further formulas for /π that arise from these alternative theories, but he provides no details for his proofs In an appendix at the end of Ramanujan s Collected Papers, [RamanujanCP, p 6], the editors, quoting LJ Mordell, lament It is unfortunate that Ramanujan has not developed in detail the corresponding theories referred to in Ramanujan s formulas for /π were not established until 987, when they were first proved by JM PB Borwein [Borwein], [Borwein], [BorweinAGM,pp 77 88] To prove these formulas, they needed to develop only a very small portion of the corresponding theories to which Ramanujan alluded In particular, the main ingredients in their work are Clausen s formula identities relating F, ; ; x, to each of the functions F, ; ; x, F, ; ; x, F 6, 5 6 ; ; x The Borweins [BorweinRR], [Borwein], further developed their ideas by deriving several additional formulas for /π Ramanujan s ideas were also greatly extended by DV GV Chudnovsky [Chudnovsky] who showed that other transcendental constants could be represented by similar series that an infinite class of such formulas existed Ramanujan s corresponding theories have not been heretofore developed Initial steps were taken by K Venkatachaliengar [Venkatachaliengar, pp 89 95] who examined some of the entries in Ramanujan s notebooks [Ramanujan] devoted to his alternative theories The greatest advances toward establishing Ramanujan s theories have been made by JM PB Borwein [Borwein] In searching for analogues of the classical arithmetic geometric mean of Gauss, they discovered an elegant cubic analogue Playing a central role in their work is a cubic transformation formula for F, ; ; x, which, in fact, is found on page 58 of Ramanujan s second notebook [Ramanujan], which was rediscovered by the Borweins A third major discovery by the Borweins is a beautiful surprising cubic analogue of a famous theta function identity of Jacobi for fourth powers We shall describe these findings in more detail in the sequel As alluded in the foregoing paragraphs, Ramanujan had recorded some results in his three alternative theories in his second notebook [Ramanujan] In fact, six pages, pp 57 6, are devoted to these theories These are the first six pages in the 00 unorganized pages of material that immediately follow the organized chapters in the second notebook Our objective in this paper is to establish all of these claims In proving these results, it is very clear to us that Ramanujan had established further results that he unfortunately did not record either in his notebooks other unpublished papers or in his published papers Moreover, Ramanujan s work points the way to many additional theorems in these theories, we hope that others will continue to develop Ramanujan s beautiful ideas The most important of the three alternative theories is the one arising from the hypergeometric function F, ; ; x The theories in the remaining two cases are more easily extracted from the classical theory so are of less interest We first review the terminology in the relevant classical theory of elliptic functions, which, for example, can be found in Whittaker Watson s text [WW] However, since we utilize many results in the classical theory that were initially found recorded by Ramanujan in his notebooks [Ramanujan], we employ much of his notation The complete elliptic integral of the first kind K Kk associated with the modulus k, 0 < k <

3 , is defined by RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES K : π/ 0 dφ k sin φ π F, ; ; k, where the latter representation is achieved by exping the integr in a binomial series integrating termwise For brevity, Ramanujan sets z : π K F, ; ; k The base or nome q is defined by q : e πk /K, where K Kk Ramanujan sets xor α k Let n denote a fixed positive integer, suppose that n F, ; ; k F, ; ; k F, ; ; l F,, ; ; l where 0 < k, l < Then a modular equation of degree n is a relation between the moduli k l which is implied by Following Ramanujan, we put α k β l We often say that β has degree n, or degree n over α The multiplier m is defined by 5 m F, ; ; α F, ; ; β We employ analogous notation for the three alternative systems The classical terminology described above is represented by the case r below For r,,, 6 0 < x <, set 6 zr : zr; x : F r, r ; ; x r In particular, q r : q r x : exp π cscπ/r F r, r r ; ; x F r, r r ; ; x q exp π F, ; ; x F,, ; ; x q exp π F, ; ; x F,, ; ; x q 6 exp π F 6, 5 6 ; ; x F 6, 5 6 ; ; x

4 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN We consider the notation 7 9 to be more natural than that of Ramanujan quoted at the beginning of this paper Let n denote a fixed natural number, assume that 0 n F r, r r ; ; α F r, r r ; ; α F r, r r ; ; β F r, r, r ; ; β where r,,, or 6 Then a modular equation of degree n is a relation between α β induced by 0 The multiplier mr is defined by mr F r, r r ; ; α F r, r, r ; ; β for r,,, or 6 When the context is clear, we omit the argument r in q r, zr, mr In the sequel, we say that these theories are of signature,,, 6, respectively Theta functions are at the focal point in Ramanujan s theories His general theta function fa, b is defined by fa, b : a nn+/ b nn /, ab < n If we set a qe iz, b qe iz, q e πiτ, where z is an arbitrary complex number Imτ > 0, then fa, b ϑ z, τ, in the classical notation of Whittaker Watson [WW, p 6] In particular, we utilize three special cases of fa, b, namely, ϕq : fq, q q n, ψq : fq, q n q nn+/, n0 f q : f q, q n n q nn+/ q n, where q < One of the fundamental results in the theory of elliptic functions is the inversion formula [WW, p 500], [BerndtIII, p 0, eq 6] n 5 z F, ; ; x ϕ q We set 6 z n ϕ q n, for each positive integer n, so that z z Thus, by 5, 5, 6, 7 m z z n ϕ q ϕ q n

5 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 5 In the sequel, unattended page numbers, particularly after the statements of theorems, refer to the pagination of the Tata Institute s publication of Ramanujan s second notebook [Ramanujan] We employ many results from Ramanujan s second notebook in our proofs, in particular, from Chapters 7, 9, 0, Proofs of all of the theorems from Chapters 6 of Ramanujan s second notebook can be found in [BerndtIII] Ramanujan s Cubic Transformation, the Borweins Cubic Theta-Function Identity, the Inversion Formula In classical notation, the identity ϑ q ϑ q + ϑ q is Jacobi s famous identity for fourth powers of theta functions In Ramanujan s notation, this identity has the form [Ramanujan], [BerndtIII, Chapter 6, Entry 5vii] ϕ q ϕ q + 6qψ q JM PB Borwein [Borwein] discovered an elegant cubic analogue which we now relate For ω expπi/, let aq : q m +mn+n, m,n bq : m,n ω m n q m +mn+n, cq : q m+/ +m+/n+/+n+/ m,n Then the Borweins [Borwein] proved that 5 a q b q + c q They also established the alternative representations 6 aq + 6 q n+ qn+ qn+ q n+ n0 7 aq ϕqϕq + qψq ψq 6 Formula 6 can also be found in one of Ramanujan s letters to Hardy, written from the nursing home, Fitzroy House [RamanujanLN, p 9], is proved by one of us in [Berndt] The identity 7 is found on page 8 in the unorganized portions of Ramanujan s second notebook [Ramanujan], [BerndtIV, Chapter 5, Entry 7] was proved by one of us [BerndtIII, Chapter, eq

6 6 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN 6] in the course of proving some related identities in Section of Chapter in Ramanujan s second notebook [Ramanujan] Furthermore, the Borweins [Borwein] proved that 8 bq { aq aq } 9 cq {aq aq } The Borweins proof of 5 employs the theory of modular forms on the group generated by the transformations t /t t t + i Shortly thereafter, JM Borwein, PB Borwein, F Garvan [BBG] gave a simpler, more elementary proof that does not depend upon the theory of modular forms Although Ramanujan does not state 5 in his notebooks, we shall show that 5 may be simply derived from results given by him in his notebooks Our proof also does not utilize the theory of modular forms We first establish parametric representations for aq, bq, cq Lemma Let m z /z, as in 7 Then 0 aq m + 6m z z, m bq z z m9 m m cq z z m + m m, Proof From Entry iii of Chapter 7 in Ramanujan s second notebook [Ramanujan], [BerndtIII, p ], ψq z α/q ψq 6 z β/q, where β has degree over α In proving Ramanujan s modular equations of degree in Section 5 of Chapter 9 of Ramanujan s second notebook, Berndt [BerndtIII, p, eq 5] derived the parametric representations α m + m 6m 5 β m + m 6m Thus, by 6, 7,,, 5, aq } z z { + αβ { } m m + z z + m m + 6m z z, m

7 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 7 so 0 is established In fact, 0 is proved in [BerndtIII, p 6, eq 5] Next, from 7 8, 6 bq ϕqϕq ϕq9 ϕq qψq ψq 6 q ψq8 ψq, from 7 9, 7 cq ϕqϕq ϕq ϕq qψq ψq 6 ψq q ψq 6 By Entry iii of Chapter 0 [Ramanujan], [BerndtIII, p 5], 6, 7, 8 ϕq9 ϕq 9 ϕ q ϕ q 9 m 9 ϕq ϕ ϕq q ϕ q m By Entry ii in Chapter 0 [Ramanujan], [BerndtIII, p 5] 5, 0 q ψq8 ψq 9q ψ q 6 ψ q 9 m β α ψq q ψq 6 ψ q q ψ q 6 m α β m m m + m + m Using 8 then 5 in 6 7, we deduce that, respectively, { bq } 9 z z m m m αβ m + z z { 9 m m z z m9 m m m m + m m m m + }

8 8 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN cq z z { z z { m + αβ m + m m + m m + m + mm m m + z z m Hence, have been established Theorem The cubic theta function identity 5 holds Proof From, b q + c q z z 6m { m m 9 m + 7m + m } z z 6m by 0 This completes the proof m + 6m a q, Our next task is to state a generalization of Ramanujan s beautiful cubic transformation for F, ; ; x E Goursat [Goursat] derived several cubic transformations for hypergeometric series, but Ramanujan s cubic transformation cannot be deduced from Goursat s results Theorem For x sufficiently small, F c, c + ; c + x ; + x } + x c F } c, c + ; c + 5 ; x 6 Proof Using MAPLE, we have shown that both sides of are solutions of the differential equation x x + x + x + x y + x[x c + x + + 8cx]y 6cc + x y 0 This equation has a regular singular point at x 0, the roots of the associated indicial equation are 0 c / Thus, in general, to verify that holds, we must show that the values at x 0 of the functions their first derivatives on each side are equal These values are easily seen to be equal, so the proof is complete Corollary p 58 For x sufficiently small, F, x ; ; + x F + x, ; ; x Proof Set c in Theorem The Borweins [Borwein] deduced Corollary in connection with their cubic analogue of the arithmetic geometric mean Neither their proof nor our proof is completely satisfactory, it would be desirable to have a more natural proof Our next goal is to prove a cubic analogue of 5 We accomplish this through a series of lemmas

9 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 9 Lemma 5 If n m, where m is a positive integer, then F, ; ; b q a aq q aq n F Proof Replacing x by x/ + x in, we find that 5 F, ; ; x + x F, ; ; b q n a q n, x ; ; + x Setting x bq/aq employing 8 9, we deduce that F, ; ; b q aq a q aq + bq F, aq bq ; ; aq + bq aq aq F, ; ; c q a q aq aq F, ; ; b q a q, by Theorem Iterating this identity m times, we complete the proof of The next result is the Borweins [Borwein] form of the cubic inversion formula Lemma 6 We have 6 F, ; ; c q a q aq Proof Letting m tend to in, noting that, by or 6 or 8, respectively, lim n aqn lim n bqn, invoking Theorem, we deduce 6 at once Lemma 7 If n m, where m is a positive integer, then 7 F, ; ; b q a aq q naq n F, ; ; b q n a q n Proof By Theorem, 5 with x cq/aq, 8, 9, F, ; ; b q a F q, ; ; c q a q aq aq + cq F, aq cq ; ; aq + cq aq aq F, ; ; b q 8 a q Replacing q by q in 8, then iterating the resulting equality a total of m times, we deduce 7 to complete the proof

10 0 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Lemma 8 Let q be defined by 7, put F x q Let n m, where m is a positive integer Then 9 F b q b a F n q n q a q n c 0 F n q a F q c q n a q n Proof Dividing by 7, we deduce that F, ; ; b q a q F, n ; ; b q a q F, ; ; b q n a q n F, ; ; b q n a q n Multiplying both sides of by π/ then taking the exponential of each side, we obtain 9 Multiply both sides of by /πn, take the reciprocal of each side, use Theorem, then take the exponential of each side We then arrive at 0 We now establish another fundamental inversion formula Lemma 9 Let F be defined as in Lemma 8 Then F c q a q q Proof Letting n tend to in 0 employing Example in Section 7 of Chapter in Ramanujan s second notebook [BerndtII, p 8], we find that F c q a lim q F /n n lim n lim n q, c q n a q n c q n 7a q n q n + where in the penultimate line we used c q n 9 a q n qn + + Theorem 0 p 58 Let F be defined as in Lemma 8 Then /n /n z : F, ; ; x af x aq

11 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES Proof Let u ux b F x/a F x Then by Lemma 6 Theorem, af x F, ; ; u On the other h, by Lemma 9, F u F x, or F, ; ; u F, ; ; u F, ; ; x ; ; x F, By the monotonicity of F, ; ; x on 0,, it follows that, for 0 < x <, 5 F, ; ; u F, ; ; x The argument is given in more complete detail in [BerndtIII, p 0] with F, ; ; x replaced by F, ; ; x In conclusion, now follows from 5 Theorem 0 is an analogue of the classical theorem, 5 Our proof followed along lines similar to those in Ramanujan s development of the classical theory, which is presented in [BerndtIII, Chapter 7, Sections 6] Corollary p 58 If z is defined by q is defined by 7, then 6 z + n where χ denotes the principal character modulo χ nnq n q n, Proof In [BerndtIII, p 60, Entry i], it is shown that 7 + n nq n q n 6 nq n q n a q n Since here q is arbitrary, we may replace q by q in 7 Theorem 0 7 Thus, 6 can be deduced from We conclude this section by offering three additional formulas for z Theorem p 57 Let q q z z Then z + 6 n q n + q n + q n

12 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Proof By Theorem 0 6, the proof is complete q n+ qn+ z + 6 qn+ q n+ n0 + 6 q n+m q n+m n0 m q m q m m m m q m q m q m n0 q m + q m + q m, In the middle of page 58, Ramanujan offers two representations for z, but one of them involves an unidentified parameter p If q is replaced by q below, then the parameter p becomes identical to the parameter p in Lemma 55 below, as can be seen from 5 Theorem p 58 Let z q be as given above Put p m /, where m is the multiplier of degree in the classical sense Then q mn 8 z ϕ q ϕq + p + p ψ q ψq 6 ϕ q ϕq Proof Our proofs will be effected in the classical base q We first assume that the second equality holds then solve it for p Let β have degree By, 9 + p + p ψ q ψq 6 m α/q β/q ϕq ϕ q m + m m, by 5 Solving 9 for p, we easily find that p m /, as claimed Secondly, we prove the first equality in 8 By the same reasoning as used in 9, ψ q ψq 6 q ϕ ϕq α z z β z + m z z m m m m + 6m z z m z aq, by 0 Appealing to Theorem 0, we complete the proof

13 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES The Principles of Triplication Trimidiation In Sections, for brevity, we set q q, z z; x unless otherwise stated In the classical theory of elliptic functions, the processes of duplication dimidiation, which rest upon Len s transformation, are very useful in obtaining formulas from previously derived formulas when q is replaced by q or q, respectively These procedures are described in detail in [BerndtIII, Chapter 7, Section ], where many applications are given We now show that Ramanujan s cubic transformation, Corollary, can be employed to devise the new processes of triplication trimidiation Theorem Let x, q q qx, z; x z be as given in 7 6, respectively Set x t Suppose that a relation of the form Ωt, q, z 0 holds Then we have the triplication formula { Ω t + t }, q, { } + t z 0 the trimidiation formula Ω t, q, + tz 0 + t Proof Set t t + t Therefore, 5 t t + t By Corollary, 6 z : zt F, ; ; t F, t ; ; + t + t F, ; ; t + tzt

14 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Also, by 7,, 5,, q : qt exp π F, ; ; t F, ; ; t exp π F, ; ; t +t F, ; ; t +t exp π F, ; ; t F, ; ; t 7 q t Thus, suppose that holds Then by 5 7, we obtain, but with t, q, z replaced by t, q, z, respectively On the other h, suppose that holds with t, q, z replaced by t, q, z, respectively Then by, 6, 7, it follows that holds Corollary With q z as above, bq x z cq x z Proof By 8, Theorem 0, the process of triplication, bq { } + x z z x z, while by 9, Theorem 0, the process of trimidiation, cq + x z z x z Theorem Recall that f q is defined by Then for any base q, 8 qf q 7 b9 qc q Proof All calculations below pertain to the classical base q By Entry ii in Chapter 7 of Ramanujan s second notebook [Ramanujan], [BerndtIII, p ], 9 qf q 6 z α α It thus suffices to show that the right side of 8 is equal to the right side of 9 To do this, we use the parameterizations for bq cq given by, respectively It will then be

15 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 5 necessary to express the functions of m arising in in terms of α β In addition to 5, we need the parameterizations 0 α m + m 6m β m + m, 6m given in 55 of Chapter 9 of [BerndtIII, p ] Direct calculations yield 9 m m α 8 α 8, β 8 β 8 m m α 8, β 8 m + β 8, α 8 5 m β 8 β 8 α 8 α 8 Hence, by,,, bq z m α 8 m m 7 6 β 8 α 8 α 8 β β 6 z m α 8 α β β 6 By,, 5, cq z β 8 β 8 β 8 m α 8 α α 7 z β 8 β m α α 6 It now follows easily from 6 7 that which, by 9, completes the proof 7 b9 qc q 6 z α α,

16 6 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Corollary p 57 Let q q, let z be as in Theorem 0 Then 8 q f q z 8 x x 8 Proof By Theorem Corollary, from which 8 follows qf q 7 x xz, Corollary 5 p 57 With the same notation as in Corollary, q 8 f q z 8 x 8 x Proof Applying to 8 the process of triplication enunciated in Theorem, we deduce that q 8 f q z as desired z 8 { } + x 5 8 { x + x { } x { 8 x z 8 { x} 8 x z 8 x x 8, } 8 + x x + x + x + x The Eisenstein Series L, M, N We now recall Ramanujan s definitions of L, M, N, first defined in Chapter 5 of his second notebook [Ramanujan], [BerndtII, p 8] thoroughly studied by him, especially in his paper [Ramanujan], [RamanujanCP, pp 6 6], where the notations P, Q, R are used instead of L, M, N, respectively Thus, Lq : Mq : + 0 Nq : 50 n n n nq n q n, n q n q n, n 5 q n q n We first derive an analogue of Entry 9iv in Chapter 7 in Ramanujan s second notebook [Ramanujan], [BerndtIII, p 0] } 8 8

17 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 7 Lemma Let q q be defined by 7, let z be as in Theorem 0 Then Lq xz + x xz dz dx Proof By logarithmic differentiation, On the other h, by Corollary, q d dq log qf q q d dq log q Lq n q n n nq n q n q d dq log qf q q d dq log 7 z x x q d dx dx log 7 z x x dq Now by Entry 0 in Chapter of Ramanujan s second notebook [Ramanujan], [BerndtII, p 87], Thus, d π F, ; ; x dx F, ; ; x x xz dq dx q x xz Using in, we deduce that 5 q d dq log qf q x xz z dz dx + x x x xz dz dx + xz Combining 5, we arrive at to complete the proof Theorem p 57 We have 6 Mq z + 8x Proof From [Ramanujan], [RamanujanCP, p 0] or from [BerndtII, p 0, Entry ], q dl dq { L q Mq }

18 8 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Thus, by Lemma, Mq L q x xz dl dx x z + x x xz dz dx + x x z x xz z + xz dz + x x dx where we have employed the differential equation for z [Bailey, p ] { d 7 x x dz } dx dx 9 z Upon simplifying the equality above, we deduce 6 Theorem p 57 We have 8 Nq z 6 0x 8x dz dx dz dx + z 9 z Proof From [Ramanujan], [RamanujanCP, p ] or from [BerndtII, p 0, Entry ], Thus, by, Lemma, Theorem, so 8 has been proved Theorem p 57 We have q dm dq {LqMq Nq} Nq LqMq x xz dm { dx xz + x xz dz } z + 8x dx { x xz z + 8x dz } dx + 8z z 6 { x + 8x x x} z 6 0x 8x, Mq z 8 9 x Proof Apply the process of triplication to 6 Thus, by Theorem, Mq { } 8 z + x x x the proof is complete 9 z + x 9 z + 8 x 9 z 9 8x, x + x,

19 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 9 Theorem 5 p 57 We have Nq z 6 x x Proof Applying the process of triplication to 8, we find that { Nq z x x 0 + x z6 6 } { } 6 x 8 + x { 6 } 6 + x 0 + x x 8 x z6 6 { x + 6 x } z x + 6x z 6 x x We complete this section by offering a remarkable formula for z an identity involving the 6th powers of the Borweins cubic theta functions bq cq Corollary 6 We have 9 0z 9Mq + Mq Proof Using Theorems on the right side of 9, we easily establish the desired result Corollary 7 We have 0 8 { b 6 q c 6 q } 7Nq + Nq Proof Using Theorems 5 on the right side of 0, also employing Corollary, we readily deduce 0 5 A Hypergeometric Transformation Associated Transfer Principle We shall prove a new transformation formula relating the hypergeometric functions z z employ it to establish a means for transforming formulas in the theory of signature to that in signature, conversely We first need to establish several formulas relating the functions ϕq, ψq, f q with aq, bq, cq Let, as customary, a; q : a aq aq, q < For any integer n, also set a; q n a; q a; q n

20 0 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN From the Jacobi triple product identity [BerndtIII, pp 6,7], 5 5 ϕ q q; q q; q, ψq q ; q q; q, 5 f q q; q Lemma 5 We have bq f q f q, cq q cq c q f q f q, ϕ q ϕ q, c q cq q ψ q 6 ψq, 58 c q c q q ψ q 6 ϕ q ψq ϕ q Proof First, 5 55 follow directly from Corollaries,, 5 Next, from 55 5, cq c q q6 ; q 6 q; q q ; q q ; q 6 Using 5, we readily find that the right side above equals ϕ q/ϕ q, the proof of 56 is complete Again, from 55 5, c q cq q q ; q 6 q ; q q ; q q 6 ; q 6, which, by 5, is seen to equal q ψ q 6 /ψq Thus, 57 is proved Lastly, 58 follows from combining Lemma 5 We have Proof By 58, we want to prove that ϕ q ϕ q cq cq ϕ q ϕ q ϕ q q ψ q 6 ϕ q ψq ϕ q

21 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES By Entry 0ii of Chapter 7 of the second notebook [BerndtIII, p ], 59 ϕ q z α ϕ q z β, where β has degree over α Thus, by 59, it suffices to prove that z β z α z β β Since m z /z, the last equality is equivalent to the equality, m α β β α α 8 8 α β 8 8 α β By, 5, 0,, the last equality can be written entirely in terms of m, namely, m m mm + m m + This equality is trivially verified, so the proof is complete Lemma 5 We have + ψ q qψ q 6 cq cq Proof By 58, the proposed identity is equivalent to the identity ψq qψ q 6 + ψ q ψq 6 ϕ q ϕ q By 59, the previous identity is equivalent to the identity, + m α α β β 8 β α By, 5, 0,, the last equality can be expressed completely in terms of m as + m + m mm m m + Since this last equality is trivially verified, the proof is complete Lemma 5 We have aq ϕ q ϕ q Proof By Entry i in Chapter 7 [BerndtIII, p ], 50 ψq + q ψ q ψq 8 z α/q 8 ψq z β/q 8

22 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Thus, by Entry i in Chapter of Ramanujan s second notebook [BerndtIII, p 60], 50,, 5, 0,, aq ψ q ψq + q ψ q ψq α z z β 8 + q ψ q ψq z m + m + q ψ q z ψq m + z z z z β α + m + q ψ q ψq + β α 8 + q ψ q ψq By 59 50, the right h side above equals The desired result now follows ϕ q ϕ q + q ψ q ψq Lemma 55 If p : pq : cq /cq, then + q ψ q ψq p ϕ q ϕ q, + p cq cq + p c q cq ψ q qψ q 6, c q c qcq, 5 + p + p aq cq c q Proof Equations 5 5 follow from Lemmas 5 5, respectively As observed in [BBG], it follows from the definition of cq that 55 cq + c q cq

23 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES Thus, by 55, 55, 5, + p c q cq q ; q q; q q; q q ; q Thus, 5 has been verified Lastly, by Lemma 5, 56, 57, 56 aq Hence, by 5 56, which proves 5 Theorem 56 p 58 If q ; q 6 q 6 ; q 6 q; q q; q q ; q q ; q q6 ; q q 6 ; q 6 q; q q; q q ; q 6 q ; q q ; q q ; q q6 ; q 6 9 q; q q ; q q ; q q ; q 6 q ; q c q c qcq c q cq + q c cq + p + p c q c qcq + q c c q cq c q cq aq c, q c q cq + q c cq α p + p + p β 7p + p + p + p, then, for 0 p <, 57 + p + p F, ; ; α + p F, ; ; β Proof By Lemma 55 58, 58 α 6 c q c q ϕ q ϕ q ψ q qψ q 6 6q ψ q 6 ϕ q ϕ q ϕ q,

24 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN by Jacobi s identity for fourth powers,, with q replaced by q Thus [BerndtIII, p 98, Entry ], with q replaced by q, 59 F, ; ; α ϕ q Also, by Lemma 55, Thus, by Lemma 6, β c q a q 50 F, ; ; β aq By Lemma 55, 59, 56, + p + p F, ; ; α aq cq ϕ q c q ϕ q ϕ q aq + p aq + p F, ; ; β, by 50 Lastly, we show that our proof above of 57 is valid for 0 < p < Observe that dα dp 6p + p + p dβ dp 7p p + p + p + p + p Hence, αp βp are monotonically increasing on 0, Since α0 0 β0 α β, it follows that Theorem 56 is valid for 0 < p < We now prove a corresponding theorem with α β replaced by α β, respectively Corollary 57 Let α β be as defined in Theorem 56 Then, for 0 < p, 5 + p + p F, ; ; α + 6p F, ; ; β Proof By 59 50, with q replaced by q, aq F 5 ϕ q, ; ; c q a q F, ; ; ϕ q ϕ q Thus, by 5 5, it suffices to prove that aq 5 ϕ q F, ; ; c q a q F, ; ; ϕ q ϕ q

25 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 5 By Lemma 9, 5 q exp π F, ; ; c q a q F, ; ; c q a q by Entry 5 of Chapter 7 in Ramanujan s second notebook [BerndtIII, p 00], 55 q exp π F, ; ; ϕ q ϕ q F, ; ; ϕ q Combining 5 55, we find that 56 ϕ q F, ; ; c q a q F, ; ; ϕ q ϕ q F, ; ; c q a q F, ; ; ϕ q ϕ q We now see that 5 follows from combining 5 56 Corollary 58 With α β as above, for 0 < p <, 57 F, ; ; α F, ; ; α F, ; ; β F, ; ; β Proof Divide 5 by 57 The authors first proof of Corollary 57 employed Theorem 56 a lemma arising from the hypergeometric differentials equation satisfied by F a, b; c; x F a, b; c; x We are grateful to Heng Huat Chan for providing the proof that is given above Corollary 58 is important, for from 57 7, 58 q : q β q α : q, where q qα denotes the classical base Thus, from Theorem 56 58, we can deduce the following theorem Theorem 59 Transfer Principle Suppose that we have a formula 59 Ω q α, z; αp 0 in the classical situation Then 50 Ω in the theory of signature + p qβ, z; βp 0, + p + p If the formula 59 involves α, in addition to its appearance through q z, it may be possible to convert 59 into a formula 50 involving only β, q, z This good fortune is manifest in

26 6 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN the next three instances, as we offer alternative proofs of Corollary 5, Theorem, Theorem 5 Second proof of Corollary 5 By elementary calculations, 5 α p + p + p β + p + p p + p + p From Entry iii of Chapter 7 [BerndtIII, p ], Theorem 56, 58, 50, f q f q z {α α/q} q p z p + p p + p + p + p + p z 7p + p 8 + p + p p q 8 + p + p + p + p 8 z β 8 β, by Theorem 56 5 This completes our second proof of Corollary 5 Second proof of Theorem By Entry i of Chapter 7 [BerndtIII, p 6], Theorem 56, 58, 50, Mq Mq z α + α + p + p + p z p + p + p + p6 + p + p z + p 5p + p 5 + p 6 + p + p { + p + p z 6p + p } + p + p z 89 β q 8 Second proof of Theorem 5 By Entry ii of Chapter 7 [BerndtIII, p 6], Theorem 56, 58, 50, Nq Nq z 6 + α α α z 6 + p + p + p + p 6 + p + p p + p + p p + p z 6 { + p + p + p + p 6 6 8p + p + p + p + 7p + p } z 6 β β

27 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 7 Having thus proved Corollary 5, Theorem, Theorem 5, we may use the process of trimidiation to reprove Corollary, Theorem, Theorem 6 More Higher Order Transformations for Hypergeometric Series The first theorem will be used to prove Ramanujan s modular equations of degree in the theory of signature Theorem 6 p 58 If 6 α : αp : p + p + p β : βp : p + p, then, for 0 p <, 6 F, ; ; α + p F, ; ; β Proof We first prove that 6 aq aq c q cq From Entry i,ii in Chapter of Ramanujan s second notebook [BerndtIII, p 60], 6 aq ϕ q ϕq + ϕ q ϕq Thus, by 6, Theorem, 7,,, 5, 50, 57, aq aq ψ q ψq 6 5ϕ q ϕq α z z z z β 5 z z ϕ q ϕq z z + m m m 5 m z m z z z β α 6q ψ q ψq c q cq, 8

28 8 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN which completes the proof of 6 Secondly, we prove that 65 aq + aq c q cq By 6, Theorem, 7,,, 5, 0,, 59, 56, which proves 65 Now let aq + aq ψ q ψq 6 ϕ q ϕq α z z z z + ϕ q ϕq z + z β z z + m + m z m + z z z ϕ q ϕ q c q cq, β α 66 p : pq : aq aq Note that p tends to 0 as q tends to 0, by Then by 66, 6, 65, p + p α + p aq aq aq aq + a q a q aq aq aq + aq 67 Also, by 66, 6, 65, 68 β p + p c q cq c q a q c q c q a q a q aq aq aq aq + c q c q a q c q cq c q a q

29 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 9 The desired result now follows immediately from Lemma 6, 66, 67, 68 We now determine those values of p for which our proof of 6 above holds By 6, dα dp + p p + p dβ dp p + p Thus, αp βp are monotonically increasing on 0, Since α0 0 β0 α β, it follows that 6 holds for 0 p < As functions of p, the left right sides of 6 are solutions of the differential equation, p p + p + p + pu + + p p 6p p p u p + pu Corollary 6 Let α β be defined by 6 Then, for 0 < p, 69 F, ; ; α + p F, ; ; β Proof By Lemma , respectively, 60 q exp π F, ; ; α F, ; ; α 6 q exp π F, ; ; β F, ; ; β The desired result now follows easily from 6, 60, 6 Corollary 6 Let αp βp be given by 6 Then, for 0 < p <, 6 F, ; ; α F, ; ; α F, ; ; β F, ; ; β 6 m + p, where m is the multiplier of degree for the theory of signature Proof Divide 69 by 6 Since 6 is the defining relation for a modular equation of degree in the theory of signature, 6 follows from 0 6 The next transformation is useful in establishing Ramanujan s modular equation of degree

30 0 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Theorem 6 p 58 Let 6 α : αp : Then, for 0 p <, 65 + p p F, ; ; α 7p + p + p + p β : βp : 7p + p + p p + p + p F, ; ; β Proof For brevity, set zx F, ; ; x In view of Theorem 6, we want to find x y so that y + y y + y z + y + yz x + x x + x 66 + yz + x + y + xz, 67 y + y 7p + p + y + p + p, 68 x + x 7p + p + p p, 69 + y + x + p + p + p p Solving 68 for x, or judiciously guessing the solution with the help of 69, we find that 60 x p + p p An elementary calculation shows that 68 then holds Substituting 60 into 69 solving for y, we find that 6 y p + p + p Substituting 6 into the left side of 67, we see that, indeed, 67 holds Lastly, it is easily checked that with x y as chosen above, x + x + x y + y, ie, the middle equality of 66 holds Hence, 65 is valid, the interval of validity, 0 p <, follows by an elementary argument like that in the proof of Theorem 6

31 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES Corollary 65 Let α β be defined by 6 Then, for 0 < p, 6 + p p F, ; ; α + p + p F, ; ; β Proof The proof is analogous to that for Corollary 6 Corollary 66 Let α β be defined by 6 Then, for 0 < p <, 6 F, ; ; α F, ; ; α F, ; ; β F, ; ; β Proof Divide 6 by 65 7 Modular Equations in the Theory of Signature We first show that Corollary 6 can be utilized to prove five modular equations of degree offered by Ramanujan Theorem 7 p 59 If β has degree in the theory of signature, then i αβ + { α β}, ii α β { α β } + p m, iii { β α } β α m, iv α β + { α β } m, v { β α } + β α m Proof From 6, by elementary calculations, 7 α p + p + p β Thus, from 6 7, respectively, p + p αβ p + p + p { α β} p + p + p

32 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Hence, αβ + { α β} p + p + p p + p Similarly, by 6, 7, 6, α β { α β } p + + p + p + p p + p + p m, { β α } β α + p + p p + p + p m, α β + { β α { α } + β β α } + p + p + p + p + p m, + p + p Thus, the proofs of i v have been completed + p + p + p m Theorem 7 p 59 Let β be of degree, let m be the associated multiplier in the theory of signature Then β β β β m + α α m α α Proof From 6, we easily find that 7 α p + p + p + p + p β p + p + p + p p Thus, from 6 7, 7 β α + β α p + p + p + p + p p + + p + p + p p + p p + p + p + p + p p + p p From Theorem 6, 7 m + p + p + p p Hence, by 7 7, 75 m β β + p p p + p + p + p α α + p + p + p p + p p p + p + p + p + p p + p p

33 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES Therefore, combining 7 75, we deduce that β α + β α m by 7, the proof is complete β β α α + p + p + p + p p + p p p + p + p + p + p p + p p + p + p + p p m, Theorem 7 p 0, first notebook Let α, β, γ have degrees,,, respectively Let m m denote the multipliers associated with the pairs α, β β, γ, respectively Then {β β} 6 {α γ} {γ α} m m Proof In 6, replace β by γ, so that for 0 p <, 76 α 7p + p + p + p γ 7p + p + p p From the proof of Theorem 6, β has the representations y + y x + x + x, where x y are given by 60 6, respectively In either case, a short calculation shows that 77 β 7p + p + p + p Using 76 77, we find that α p6 p 5 6p + p 6p p + + p + p p p + 5p + + p + p, β p6 + p 5 p 6p p + p + + p + p p p + 5p + + p + p, γ p6 5p 5 9p p + p + 8p p p p p + p + 5p + + p p

34 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Hence, from 76 70, 7 {β β} 6 {α γ} {γ α} 7p + p p p + 5p + + p + p + p + p 7p + p pp + p + 5p + + p + p + p p + p + p + p p + p + p { + p + p p p} + p + p + p p + p + p 6 7p + p p p + 5p + + p p + p + p On the other h, from the proof of Theorem 6, from 60 6, 7 m + y m + x + p + p + p p + p + p Combining 7 7, we complete the proof Next, we show that Ramanujan s beautiful cubic transformation in Corollary yields the defining relation for modular equations of degree We then iterate the transformation in order to derive Ramanujan s modular equation of degree 9 Lemma 7 If β 7 α, + β then 7 F, ; ; β F, ; ; β F, ; ; α ; ; α F, Furthermore, the multiplier m is equal to + β Proof In 5, set x β Dividing 5 by, we deduce 7 The formula m + β is an immediate consequence of Theorem 75 p 59 If m is the multiplier for modular equations of degree 9, then + β 75 m, + α

35 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 5 where β has degree 9 Proof Let α be given by 7, but with β replaced by Applying twice, we find that We want to express the multiplier β t : + β F, ; ; α + t F, ; ; t + t + β F, ; ; β 76 m + t + β entirely in terms of α β Solving for β in 7 then replacing β by t, we find that Thus, t α α + + t α + Using this in 76, we deduce 75 to complete the proof Theorem 76 p 59 If β has degree 5, then 77 αβ + { α β} + {αβ α β} 6 Proof By Corollary, we may rewrite 77 in the form 78 bqbq 5 + cqcq 5 + bqcqbq 5 cq 5 aqaq 5 From Theorem 5 55 of Lemma 5, we find that { f q + 7qf q } 79 aq f qf q In fact, 79 is given by Ramanujan in his second notebook [BerndtIII, p 60, Entry i] Thus, by 5, 55, 79, 78 is equivalent to the eta function identity f qf q 5 f q f q 5 + f q f q 5 9q f qf q 5 + 9qf qf q f q 5 f q 5 70 { f q + 7qf q } { f q 5 + 7q 5 f q 5 } f qf q f q 5 f q 5

36 6 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Cubing both sides of 70, simplifying, setting A f q, B f q, C f q 5, D f q 5, we deduce that 7 is equivalent to the proposed identity 5q A 6 B 6 C 6 D 6 + 0qA 8 C 8 B D + 90q A C B 8 D 8 + A 0 C 0 B D + 8q A C B 0 D 0 7 q A D + B C Setting P f q q f q Q f q5 q 5 f q 5 dividing both sides of 7 by q ABCD 6, we find that 7 can be written in the equivalent form 5 + 0P Q + 90 P Q + P Q + 8 P Q P 6 Q 6 + Q6 P 6, or 7 { P Q } P Q { Q P } P Q By examining P Q in a neighborhood of q 0, so that the proper square root can be taken on the right side of 7, we find that 7 is equivalent to the identity 7 P Q Q P P Q P Q Now 7 is stated by Ramanujan on page of his second notebook [Ramanujan] has been proved by Berndt in his book [BerndtIV, Chapter 5, Entry 6] See also a paper by Berndt L C Zhang [BZ], where several of Ramanujan s eta function identities similar to 7 are proved This therefore completes the proof of 77 Theorem 77 p 59 If β has degree 7, then 7 m β α + β α 7 m β β α α β β α α Proof Using Corollary recalling that m z /z 7, we find that 7 is equivalent to the equality 75 bq7 bq + cq7 cq cq 7 bq 7 cq 7 7bq7 bqcq bqcq Employing 5 55 in 75 then multiplying both sides of the resulting equality by f qf q / qf q 7 f q, we find that 75 may be written in the equivalent form 6 76 f qf q qf q 7 f q qf qf q f q f q 7 + f q f q 7 qf qf q 7 qf q7 f q f qf q

37 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 7 If we set P f q q f q 7 we deduce that 76 is equivalent to the identity Q f q q f q, 77 P Q + 7 P Q P Q + Q P However, 77 can be found on page of Ramanujan s second notebook [Ramanujan] has been proved by Berndt Zhang [BZ, Theorem ] See also [BerndtIV, Chapter 5, Entry 68] This completes the proof of 7 Theorem 78 p 59 If β has degree, then αβ + { α β} + 6 {αβ α β} {αβ α β} } {αβ 6 + { α β} 6 Proof Employing Corollary, we find that 78 is equivalent to the identity cqcq + bqbq + 6 bqbq cqcq 79 + { bqbq cqcq } { cqcq + bqbq } aqaq By 5, 55, 79, 79 can be transformed into the equivalent identity 9 q f q f q f qf q +9q f qf q f q f q + f qf q f q f q + 8q f qf q f q f q } {q f q f q f qf q f qf q + f q f q 70 { f q + 7qf q } { f q + 7q f q } f qf q f q f q Setting A f q, B f q, C f q, D f q, multiplying both sides of 70 by ABCD, we find that it suffices to prove that 9q B D + A C + 8q A B C D + 7q ACB D + 9qA C BD 7 { A + 7qB } { C + 7q D }

38 8 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN We next cube both sides of 7, simplify, divide both sides of the resulting equality by 7qABCD 6 After considerable algebra, we deduce that 5 AC + qbd { AC + qbd qbd AC { AC + qbd } qbd AC } qbd AC { AC qbd { AC qbd + qbd } AC } qbd + AC 6 AD 7 q 5 + BC q 5 Now set P A q B ηz ηz 6 BC AD Q C q D ηz ηz, where ηz denotes the Dedekind eta function, q expπiz, Imz > 0 Then 7 is equivalent to the identity { 5 } { } P Q P Q P Q + P Q P Q P Q 7 +5 { P Q + } { + 69 P Q + } 6 P P Q P Q Q 6 Q P The beautiful eta function identity 7 has the same shape as many eta function identities found in Ramanujan s notebooks, but is apparently not in Ramanujan s work In contrast to our proofs of most of these identities, we shall invoke the theory of modular forms to prove 7 All of the necessary theory is found in [BZ] or [BerndtIV, Chapter 5] Let Γ denote the full modular group, let MΓ, r, v denote the space of modular forms of weight r multiplier system v on Γ, where Γ is a subgroup of finite index in Γ As usual, let { a b Γ 0 N c d } Γ : c 0modN Then by [BZ, Lemma ], P Q MΓ 0, 0,, by [BZ, Lemma ], P/Q 6 MΓ 0, 0, Let ordg; z denote the invariant order of a modular form g at z Let r/s, r, s, denote a cusp for the group Γ Then, for any pair of positive integers m, n, 7 ord ηmnz; r mn, s s mn, where η denotes the Dedekind eta function

39 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 9 A complete set of cusps for Γ 0 is { 0,,, } Using 7, we calculate the orders of P Q P/Q at each finite cusp The following table summarizes these calculations: function g cusp ζ ordg; ζ P Q 0 P/Q Let L R denote the left right sides, respectively, of 7 Using the valence formula Rankin [Rankin, p 98, Theorem ] the table above, we find that 75 0 ordl; + ordl; 0 + ordl; + ordl; ordl; ordl; ordr; ordr; By 75 76, if we can show that 77 L R Oq, as q tends to 0, then we shall have completed the proof of 7 Using Mathematica, we find that L q 5 + 6q + 7q + 9q + 79q q + 56q + Oq R Thus, the proof of 77, hence also of 78, is complete At the bottom of page 59 [Ramanujan], Ramanujan records three modular equations of composite degrees Unfortunately, we have been unable to prove them by methods familiar to Ramanujan so have resorted to the theory of modular forms for our proofs It would be of considerable interest if more instructive proofs could be found Because the proofs are similar, we give the three together Theorem 79 p 59 If β, γ, are of degrees,, 8, respectively, if m m are the multipliers associated with the pairs α, β γ,, respectively, then 78 α { α } {βγ β γ} 6 m m

40 0 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN Theorem 70 p 59 If β, γ, have degrees, 7, or, 5, 0, respectively, if m m are as in the previous theorem, then α + { α } βγ + { β γ} m m Proofs of Theorems Transcribing via Corollary, we see that it suffices to prove that 70 aqaq 8 cqcq 8 bqbq 8 cq cq bq bq, 7 aqaq + cqcq + bqbq aq aq 7 + cq cq 7 + bq bq 7, 7 aqaq 0 + cqcq 0 + bqbq 0 aq aq 5 + cq cq 5 + bq bq 5 Next, employing 5, 55, 7, we translate 70 7 into the equivalent eta function identities, { f q + 7qf q } { f q 8 + 7q 8 f q } f qf q f q 8 f q 7 9q f q f q f qf q 8 f qf q 8 f q f q 9qf q f q f q 6 f q, { f q + 7qf q } { f q + 7q f q } f qf q f q f q +8q 5 f q f q f qf q + f qf q f q f q { f q + 7q f q 6 } { f q 7 + 7q 7 f q } f q f q 6 f q 7 f q 7 +8q f q 6 f q f q f q 7 + f q f q 7 f q 6 f q, { f q + 7qf q } { f q 0 + 7q 0 f q 60 } f qf q f q 0 f q 60 +8q 7 f q f q 60 f qf q 0 + f qf q 0 f q f q 60

41 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES { f q + 7q f q } { f q 5 + 7q 5 f q 5 } f q f q f q 5 f q q f q f q 5 f q f q 5 + f q f q 5 f q f q 5 Recall that if q expπiz, where Imz > 0, then ηz q f q is a modular form on Γ a b a b of weight If Γ d is odd, the multiplier system v c d η is given by [Knopp, c d p 5] a b c 76 v η ± e πi{ac d +db c+d }/, c d d where c d denotes the Legendre symbol, the plus sign is taken if c 0 or d 0, the minus sign is chosen if c < 0 d < 0 Using 76, we find that each of the four expressions in 7 each of the six expressions in both 7 75 has a multiplier system identically equal to, provided, of course, that c is divisible by,, 60, respectively Hence, both sides of 7 75 belong to MΓ 0 n,,, where n,, 60, respectively If σ denotes the number of inequivalent cusps of a fundamental region for Γ 0 n, then [Schoeneberg, p 0] σ d n φ d, n/d, where φ denotes Euler s φ function a, b denotes the greatest common divisor of a b Thus, for n,, 60, there are 8, 8, cusps, respectively Using a procedure found in Schoeneberg s book [Schoeneberg, pp 86 87], we find that { 0,,,, 6, 8,, } ; { 0,,, 6, 7,,, } ; { 0,,,, 5, 6, 0,, 5, 0, 0, } constitute complete sets of inequivalent cusps for Γ 0, Γ 0, Γ 0 60, respectively Employing 7, we calculate the order of each expression in 7 75 at each finite cusp In each instance, we find that each order is nonnegative Let F, F, F 60 denote the differences of the left right sides of 7 75, respectively Since the order of F, F, F 60 at each point of a fundamental set is nonnegative, we deduce from the valence formula that 77 rρ Γ0 n ordf n ;, n,, 60, provided that F n is not constant, where r is the weight of F n 78 ρ Γ0 n : [Γ : Γ 0n] Now Schoeneberg [Schoeneberg, p 79], 79 [Γ : Γ 0 n] n p n +, p where the product is over all primes p dividing n Thus, by 78 79, ρ Γ0, ρ Γ0 8, ρ Γ0 60 Since r in each case, by 77, 750 ordf ; 8, ordf ; 6, ordf 60 ;,

42 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN unless F, F, or F 60, respectively, is constant Using the pentagonal number theorem,, ie, f q n n q nn / q q + q 5 + q 7 q q 5 + q + Mathematica, we exped the left right sides of 7 75 about the cusp q 0 We found that the left right sides of 7 75 are equal to, respectively, 9q 9q 8q 5 + Oq 9, q + 8q 5 +8q 6 +6q 7 +5q 9 +8q 0 + 6q + 8q + 8q + 6q + 90q 5 +Oq 7, + 8q + 8q 7 + 8q 8 + 5q 9 + 6q + 6q + 6q + 8q q 6 + 6q 7 + 6q 9 + 6q + 5q + 90q + Oq 5 Thus, F Oq 9, F Oq 7, F 60 Oq 5, which contradicts 750 unless F, F, F 60 are each constant These constants are obviously equal to 0, hence 7 75 are established This completes the proofs of Theorems From 75 75, we are led to several interesting conjectures about the coefficients in the expansions of the left right sides of 7 75 We will not further pursue this here 8 The Inversion of an Analogue of Kk in Signature Theorem 8 p 57 Let q q be defined by 7, let z be defined by 6 with r For 0 φ π/, define θ θφ by 8 θz Then, for 0 θ π/, 8 φ θ + where q : e y, n φ 0 F, ; ; x sin t dt sinnθ n + coshny θ + sinnθq n n + q n + q n : Φθ, n Recall from Entry 5iii of Chapter [BerndtII, p 99] that 8 F + n, n; ; x x cos n sin x, where n is arbitrary With n 6 in 8, we see that the integral in 8 is an analogue of the incomplete integral of the first kind, which arises from the case n 0 in 8 Since F, ; ; x sin t

43 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES is a nonnegative, monotonically increasing function on [0, π/], there exists a unique inverse function φ φθ Thus, 8 gives the Fourier series of this inverse function is analogous to familiar Fourier series for the Jacobian elliptic functions [WW, pp 5 5] The function φ may therefore be considered a cubic analogue of the Jacobian functions Theorem 8 is also remindful of some new inversion formulas in the classical setting which are found on pages 8, 85, 86 in Ramanujan s second notebook [Ramanujan] which have been proved by Berndt Bhargava [BB] When φ 0 θ, 8 is trivial When φ π/, φ 0 F, ; ; x sin t dt n0 n n nn! π/ x n sin n t dt n n n0 x n n π nn! n! π F, ; ; x 0 Thus, θ π/, which is implicit in our statement of Theorem 8 We now give an outline of the proof of Theorem 8 Returning to 8, we observe that 8 Sx : F, ; ; x x cos sin x, x <, is that unique, real valued function on, satisfying the properties Sx is continuous on,, S0, 87 x S x Sx 0 Properties are obvious, 87 follows from the elementary identity cos θ cos θ + cosθ To see that Sx is unique, set y Sx in 87 solve for x Thus, x y + y y Since Sx is real valued, either y < 0 or y Since S0 S is continuous, we conclude that y Hence x ±gy, where /y y + gy : y Now, gy is monotonically increasing on [, Thus, g x exists, if 0 x <, y g x, while if < x < 0, we have y g x Thus, Sx is uniquely determined

44 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN We fix q, 0 < q < Set x c q/a q, so by 5, 0 < x < Then, by Lemma 6, Z : F, ; ; x aq We use the notation Z instead of z z, because in this section z will denote a complex variable With Φθ defined in 8, we shall prove that 88 dφθ dθ > 0, 0 < θ < π/, 89 x sin Φθ Z dφ dθ Z dφ dθ By 88, we may define Θ : Φ : [0, π/] [0, π/] Setting S : Z dθ/dφ, we see from 89 that S x sin φ S 0 Hence, 87 holds with x replaced by x sin φ Now, from 8, Φ n q n + q n Z, + qn by Theorem Thus, S0 Z/Φ 0 Hence, by 8 87, we conclude that so Z dθ dφ F, ; ; x sin φ, 0 < φ < π/, ZΘφ φ 0 F, ; ; x sin t dt, 0 φ π/, since Θ0 0 It follows that Θφ θφ that φ Φθ Hence, the desired result 8 holds Proof of Theorem 8 For q < z < / q, define 80 vz, q : + n z n + z n q n + q n + q n We note, by Theorem, that v, q aq,, by 8, that 8 V θ : ve iθ, q dφ dθ By exping / q n in a geometric series inverting the order of summation, we find that 8 vz, q + { zq n+ n0 zqn+ zqn+ zq n+ + z q n+ z q n+ z q n+ z q n+ }

45 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 5 As a function of z, vz, q can be analytically continued to C \ {0}, where the analytic continuation vz, q is analytic except for simple poles at z q m, where m is an integer such that m 0 mod Using 8, we find, by a straightforward calculation, that 8 vzq, q vz, q Hirschhorn, Garvan, J Borwein [HGB] have studied generalizations of aq, bq, cq in two variables In particular, they defined showed that [HGB, eq ] bz, q : m,n ω m n q m +mn+n z n 8 bz, q q; q q ; q zq; q; z q; q zq ; q z q ; q q n zq n z q n q n q n n n [HGB, eq 7] 85 bzq, q z q bz, q We next show that vz, q can be written in terms of bz, q b z, q Lemma 8 If αq : q; q q ; q q ; q q 6 ; q 6 then βq : q; q6 q 6 ; q 6 q ; q q ; q, 86 vz, q q αqb z, bz, q βq The case z of 86 follows from [HGB, eq 9] To prove Lemma 8, we employ the following lemma due to AOL Atkin P Swinnerton Dyer [ASD] Lemma 8 Let q, 0 < q <, be fixed Suppose that fz is an analytic function of z, except for possibly a finite number of poles, in every region, 0 < z z z If fzq Az k fz

46 6 BRUCE C BERNDT, S BHARGAVA, AND FRANK G GARVAN for some integer k positive, zero, or negative some constant A, then either fz has k more poles than zeros in the region q < z, or fz vanishes identically Proof of Lemma 8 Define 87 F z : bz, qvz, q αqb z, q + βqbz, q Examining 86, we see that our goal is to prove that F z 0 From 8 85, F zq z q F z From our previous identification of the poles of vz, q from the definition 8 of bz, q, we see that the singularities of F z are removable Thus, by Lemma 8, to show that F z 0, we need only show that F z 0 for three distinct values of z in the region q < z We choose the values z, ω, ω, where ω expπi/ For z, by 87 8, we want to prove that 88 v, q αq q; q q 6 ; q 6 q ; q q ; q βq βq But this has been proved by N Fine [Fine, p 8, eq 6] For the values z ω, ω, we need the evaluations 89 vω, q vω, q bq The first equality follows from the representation of vz, q in 8 To prove the second, we first find from 80 that v, q + vω, q + 9 q n + q n + q 6n aq, by Theorem Since v, q aq, by Theorem, we deduce that n vω, q aq aq By 8, we conclude that bq vω, q to complete the proof of 89 Now setting z ω, we see from 87 that we are required to prove that bq q αqb ω, bω, q βq 80 q; q q 6 ; q 6 q 9 ; q 9 q ; q q ; q q 8 ; q 8 βq ϕ q ϕ q ϕ q 9 ϕ q, where we employed 8, much simplification, 5 Ramanujan s second notebook [BerndtIII, p 5], ϕ q9 ϕ q 9 ϕ q ϕ q From Entry iii in Chapter 0 of

47 RAMANUJAN S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 7 Thus, by 59, 7, 0,, ϕ q ϕ q ϕ q9 ϕ q 8 ϕ q ϕ q 9 ϕ q z α z β ϕ q 9 m β α z m 8z m m z z m + m m bq, 9 m m m + m by See also [BorweinAGM, p, Theorem b] Thus, 80 has been proved Note that in 8 α β are squares of moduli are not to be confused with the definitions of αq βq in Lemma 8 In conclusion, we have shown that F z 0 for z, ω, ω, so the proof of Lemma 8 is complete We shall need some further relations among aq, cq, αq, βq First, from 8, b q ϕ9 q 8ϕ q 9 ϕ q ϕ q 8 β 9α β Since, from 8 86, with z, 8 aq α β β, it follows from 5 that 8 c q 7α 8β α β Recall from8 that V θ dφ/dθ, where Φθ is defined by 8 Our next task is to derive an infinite product representation for dv/dθ To do this, we employ Bailey s 6 ψ 6 summation [GR, p 9] Lemma 8 Let, for z <, [ ] a,, a m : a ; q a m ; q b,, b n b ; q b n ; q [ ] a,, a 6 6ψ 6 ; q; z : b,, b 6 n a ; q n a 6 ; q n b ; q n b 6 ; q n z n Then, for a q/bcde <, [ q a, q ] a, b, c, d, e 6ψ 6 ; q; a q a, a, qa/b, qa/c, qa/d, qa/e bcde 8 [ ] aq, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de, q, q/a q/b, q/c, q/d, q/e, aq/b, aq/c, aq/d, aq/e, a q/bcde