2 BRUCE C. BERNDT In Section 5 the focus is on q-continued fractions in particular those which can be represented by a q-product such as the RogersíRa

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1 FLOWERS WHICH WE CANNOT YET SEE GROWING IN RAMANUJAN'S GARDEN OF HYPERGEOMETRIC SERIES ELLIPTIC FUNCTIONS AND q's BRUCE C. BERNDT Abstract. Many of Ramanujan's ideas and theorems spawn questions and problems many of which remain unresolved or even to be thoroughly examined. This survey raises questions arising from Ramanujan's work on thetafunctions and other q-series with Gaussian hypergeometric functions making frequent appearances.. Introduction In his lecture on June 987 at the Ramanujan centenary conference at the University of Illinois Freeman Dyson ë32ë proclaimed ëthat was the wonderful thing about Ramanujan. He discovered so much and yet he left so much more in his garden for other people to discover." Our goal in this paper is to tell you about some of the æowers in Ramanujan's garden that have yet to be discovered. Of course because we have not yet seen the æowers we cannot describe their colors or fragrences for you. We cannot even identify their genus and species and in fact we may need to concoct some new botanical names. The æowers may also be extremely diæcult to ænd. What we can do is point out some signposts and provide a map with some pathways some of which may lead you nowhere. Many of our queries originate with Ramanujan's alternative theories of elliptic functions and so we begin our stroll through Ramanujan's garden basking in the beauty of these æowers the seeds of which were planted in Ramanujan's famous paper on modular equations and approximations to ë48ë ë49 pp. 23í39ë in 94 but which only germinated in his notebooks ë50ë. These æowers were not seen by anyone but Ramanujan until Berndt S. Bhargava and F. G. Garvan ëë cultivated them in 995. Since then other gardeners have found related species growing but Ramanujan has left a huge plot of land for still undiscovered related species to grow and be nurtured. Arising out of both the classical and alternative theories are transformations between hypergeometric functions. In many cases in particular with those transformations arising from modular equations we ask if these are isolated or are they really special cases of more general transformations yet to be discovered. Sections 2 and 3 give examples. In Section 4 we turn our attention to the explicit evaluation of theta-functions hypergeometric series and elliptic integrals. An important and useful analogue of the classical class invariants of Weber and Ramanujan found in Ramanujan's lost notebook is discussed. This points the way to other analogues which have not yet sprung up in Ramanujan's garden.

2 2 BRUCE C. BERNDT In Section 5 the focus is on q-continued fractions in particular those which can be represented by a q-product such as the RogersíRamanujan continued fraction which has an extensive elegant theory. Less extensive theories have been developed for the RamanujaníGordoníGíollnitz and cubic continued fractions. There appear to be many garden paths needing development. In the last garden plot the æowers are really mysterious. In his lost notebook ë5ë Ramanujan left a very beautiful rare species of æower so rare that except for a few additional æowers found by Berndt Chan and S.íS. Huang ë4ë no one has found any others or really explained how they grow or if they have relationships with other species. Ramanujan found several integrals of eta-functions that can be represented as incomplete elliptic integrals of the ærst kind. If you have not previously seen this species you will deænitely agree that it is an exotic one. 2. Ramanujan's Alternative Theories of Elliptic Functions In his famous paper ë48ë ë49 pp. 23í39ë Ramanujan records several elegant series for = and asserts ëthere are corresponding theories in which q is replaced by one or other of the functions" q r = q r x = exp csc=r 2 F! r ; r r ;; xæ 2F r ; r;;xæ ; 2. r where r =3; 4 or 6 and where 2 F denotes the classical Gaussian hypergeometric function. In the classical theory of elliptic functions the variable q is given by q 2 and Ramanujan implies that most of his series for = arise not out of the classical theory but out of new alternative theories wherein q is replaced by either q 3 ;q 4 or q 6. Ramanujan gives no proofs of his series for = or of any of his theorems in the ëcorresponding" or ëalternative" theories. It was not until 987 that J. M. and P. B. Borwein ë2ë proved Ramanujan's series formulas for =. In his second notebook ë50 pp. 257í262ë Ramanujan records without proofs his theorems in these new theories which were ærst proved in 995 by Berndt Bhargava and Garvan ëë who gave these theories the appellation the theories of signature r r = 3; 4; 6. An account of this work may also be found in Berndt's book ë9 Chap. 33ë. Deæne the complete elliptic integral Kk of the ærst kind by K = Kk = Z =2 0 d p = k2 sin F 2 ; 2 ;;k2 ; 2.2 where the latter representation is achieved by expanding the integrand in a binomial series and integrating termwise. Here k; 0 éké; is called the modulus. In the classical theory the theta-functions 'q = X n= q n2 and q = X n=0 q nn+=2 play key roles. In particular Jacobi's identity ë7 p. 40 Entry 25viië ' 4 q ' 4 q =6q 4 q is crucially used in establishing the fundamental inversion formula ë7 pp. 00í0ë z 2 = 2 F 2 ; 2 ;;x = 2 Kk ='2 q; 2.5 where x = k 2 ;q= q 2 is given by 2. and Kk is given by 2.2.

3 RAMANUJAN'S GARDEN 3 A common sumptuous æower in the classical theory is the Dedekind eta-function ; deæned by =e 2i=24 q; q = q =24 fq; q = e 2i ; 2.6 where f q is the notation employed by Ramanujan. We close our brief description of the classical theory by noting the well-known formulas ë7 p. 39 Entries 24 ii ivë and 'q = f 2 q fq 2 and q = f 2 q 2 fq In the cubic theory or the theory of signature 3 for! = exp2i=3 deæne aq = bq = cq = X q m2 +mn+n 2 ; m;n= X m;n= X ! mn q m2 +mn+n 2 ; 2.9 q m+=32 +m+=3n+=3+n+=3 2 m;n= 2.0 The functions deæned in 2.8í2.0 are the ëcubic" theta-functions ærst introduced by the Borweins ë22ë who proved that a 3 q =b 3 q+c 3 q Ramanujan ë50 p. 258ë established the fundamental inversion formula 2. z 3 = 2 F 3 ; 2 3 ;;x = aq; 2.2 where q = q 3 is given by 2.. This theorem was ærst proved in print by Berndt Bhargava and Garvan ëë ë9 p. 99ë with 2. being a necessary ingredient in their proof. The analogues of 2.7 in the theory of signature 3 are given by ë9 p. 09 Lemma 5.ë bq = f 3 q fq 3 and cq =3q =3 f 3 q 3 fq 2.3 In the theory of signature 4 or in the quartic theory Berndt Bhargava and Garvan ëë ë9 p. 46 eq. 9.7ë established a ëtransfer" principle of Ramanujan by which formulas in the theory of signature 4 can be derived from formulas in the classical theory. Taking the place of aq;bq and cq in the cubic theory are the functions and Aq =' 4 q+6q 4 q 2 ; Bq =' 4 q 6q 4 q 2 ; 2.4 Cq =8 p q' 2 q 2 q 2 ; 2.5

4 4 BRUCE C. BERNDT where ' and are deæned in 2.3 which by Jacobi's identity 2.4 satisfy the equality A 2 q =B 2 q+c 2 q; 2.6 the quartic analogue of 2.4 and 2.. Berndt Chan and Liaw ë6ë used 2.6 to establish the inversion formula z 4 = 2 F 4 ; 3 4 ;;x = p Aq; 2.7 which is clearly an analogue of 2.5 and 2.2. They therefore were able to bypass the aforementioned transfer theorem and directly prove theorems in the quartic theory. To again demonstrate the ubiquitous role of the Dedekind eta-function we note that the quartic analogues of 2.7 and 2.3 are given by ë6 Thm. 3.ë Bq = f 2 q fq 2 4 and Cq =8 p q f 2 q 2 fq Ramanujan's theory of elliptic functions of signature 6 is not as complete as those in the cubic and quartic theories ëë ë9 pp. 6í64ë. In particular we have been unable to obtain a sextic analogue of and 2.7. The few results in the sextic theory recorded by Ramanujan in his notebooks were proved by using the transformation formula p +2p 2 F 6 ; 5 6 ;;æ =p +p + p 2 2F 2 ; 2 ;;æ; where 0 épé and æ and æ are given by p2 + p æ = +2p and æ = 27p2 + p p + p 2 3 Chan and Y. L. Ong ë30ë have established a few results pointing toward the beginnings of a theory of signature 7. The analogue of 'q and aq in the septic theory is given by X q m2 +mn+2n 2 m;n= However we do not know any other theta functions in the septic theory. No further alternative theories have been found. It may be that if other theories of signature n exist then the class number of Q p n must be equal to. Observe that the discriminant ofm 2 +mn+2n 2 is 7 and that the class number of Q p 7 equals. The alternative cubic theory appears to be the most interesting of the three known alternative theories for several reasons. In particular the new theta functions aq; bq; and cq arise and the many results found by Ramanujan in the cubic theory do not seem to be reachable through the classical theory. If there is a complete cubic theory of elliptic functions analogous to the rich and beautiful classical theory then it is necessary to seek the identities of the cubic analogues of the Jacobian elliptic functions and the cubic analogues of the elliptic integrals from which the Jacobian functions arise by inverting the elliptic integrals. Ramanujan bequeathed to us one extraordinarily beautiful æower found on page 257 in his second notebook ë50ë ë Theorem 8.ë ë9 p. 33ë given below. Proofs have been given by Berndt Bhargava and Garvan ëë and by L.íC. Shen ë57ë.

5 RAMANUJAN'S GARDEN 5 Theorem 2.. Let q = q 3 be deæned by2. and let z = z 3 be deæned by2.2. For 0 ' =2; deæne = ' by Then for 0 =2; ' = +3 X n= where q = e y Z ' z = 0 2F 3 ; 2 3 ; 2 ; x sin2 t æ dt sin2n X n + 2 coshny = +3 sin2nq n n + q n + q 2n = æ; 2.20 n= Recall that ë6 p. 99 Entry 35iii Chap. ë 2.9 2F 2 + n; 2 n; 2 ; x2æ = x 2 =2 cos 2n sin x æ ; 2.2 where n is arbitrary. With n = 6 in 2.2 we see that the integral in 2.9 is an analogue of the incomplete integral of the æ ærst kind which arises from the case n = 0 in 2.2. Since 2 F 3 ; 2 3 ; 2 ; x sin2 t is a nonnegative monotonically increasing function on ë0;=2ë; there exists a unique inverse function ' = ' Thus 2.20 gives the ëfourier series" of this inverse function and is analogous to familiar Fourier series for the Jacobian elliptic functions Whittaker and Watson ë65 pp. 5í52ë. The function æ may therefore be considered a cubic analogue of the Jacobian function sn. When ' =0=; 2.20 is trivial. When ' = =2; Z ' X 0 2F 3 ; 2 3 ; 2 ; x sin2 t æ dt = = n=0 X n=0 3 n 2 3 n 2 nn! Z =2 x n sin 2n tdt 0 3 n 2 3 n 2 x n 2 n nn! n! 2 = 2 2F 3 ; 2 3 ;;xæ Thus = =2; which is implicit in our statement of Theorem 2.. What are the cubic analogues of the other Jacobian elliptic functions if they do indeed exist? Are there further integrals like 2.9 which when inverted yield these new cubic Jacobian functions? Some progress in these directions has been made in unpublished work by Y.íS. Choi. In the classical theory Pfaæ's familiar quadratic transformation ë6 p. 93ë 2F 2 ; 2 ;; x +x 2! =+x 2 F 2 ; 2 ;;x is the key to proving the inversion formula 2.5 while in the cubic theory Ramanujan's cubic transformation ë22ë ëë ë9 p. 97 Cor. 2.4ë 2F 3 ; 2 3 ;; x +2x 3! =+2x 2 F 3 ; 23 ;;x plays the leading role in proving 2.2. The ærst proof of 2.23 was by the Borweins ë22ë while another proof is given in ëë ë9 p. 97 Cor. 2.4ë. However Chan's ë25ë

6 6 BRUCE C. BERNDT two proofs are more natural. In the quartic theory the transformation ë50 p. 260ë ë9 p. 46 Thm. 9.4ë ë6 Thm. 2.ë 2F 4 ; 3 4 ;; x +3x 2! = p +3x 2 F 4 ; 34 ;;x is central. Are there any further transformations of the types 2.22í2.24? There are essentially but two quadratic transformations for 2 F ë3 p. 25 Thm. 3..; p. 27 Thm. 3..3ë; 2.22 is a special case of one of these. The transformation 2.23 is a special case of the cubic transformation ëë ë9 p. 96 p. 96ë 2F c; c + 3! 3c + x ; ; =+2x 3c 2F c; c + 3c +5 ; ; x x Further cubic transformations were found by Ramanujan ë50ë Berndt Bhargava and Garvan ëë ë9 pp. 73í75ë and Garvan ë34ë. E. Goursat ë35ë found many cubic transformations. How many distinct cubic transformations exist? Apparently no general transformations of order exceeding three have been found. We shall return to this question in the next section. 3. Modular Equations and Hypergeometric Transformations We begin by reviewing the deænition of a modular equation as it was understood by Ramanujan. In the sequel for brevity set q = q 2 and z = z 2 Let n denote a æxed positive integer and suppose that ;; k2æ ;; `2æ n 2 F 2 ; 2 2F 2 ; 2 ;;k2æ = 2 F 2 ; 2 2F 2 ; 2 ;;`2æ ; 3. where 0 ék;`é Then a modular equation of degree n is a relation between the moduli k and ` which isimplied by 3.. Following Ramanujan we put æ = k 2 and æ = `2 We often say that æ has degree n; or degree n over æ The multiplier m is deæned by m = 2 F 2 ; 2 ;;ææ 2F 2 ; 2 ;;ææ 3.2 The fundamental relations 2.5 and 3.2 are the keys to deriving modular equations for the most successful approach has been to establish theta-function identities and then convert them to modular equations by using a catalogue of formulas for theta-functions based on 2.5 and in terms of z; q; and x or æ or k In view of 2. we could also deæne modular equations in the alternative theories. Indeed Ramanujan did this and derived several modular equations in the cubic and quartic theories ëë ë9 pp. 20í3253í6ë. Further modular equations in the cubic theory were found by Chan and Liaw ë28ë. Ramanujan discovered several hundred modular equations and although almost all of them have now been proved for most of them we have not discerned Ramanujan's methods. In particular to prove some of them we have had to resort to the theory of modular forms a subject with which Ramanujan was apparently unfamiliar. See ë7ëíë9ë for proofs. With the publication of his lost notebook ë5ë there appears a fragment in which Ramanujan summarizes by type many of his

7 RAMANUJAN'S GARDEN 7 results on modular equations. Proofs for most of them can be found in ë9ë but some do not appear in his notebooks; the entire fragment is discussed in ë0ë ë4 Chap. 9ë. In this brief section we focus our attention on Ramanujan's formulas for the multiplier m; deæned in 3.2 which are conveniently gathered together in the aforementioned fragment. Almost all of Ramanujan's formulas of this sort are completely new and moreover we do not know how Ramanujan derived them. Most of our proofs use the theory of modular forms. Note that by 3.2 a formula for the multiplier yields a transformation formula for hypergeometric functions. For né3; these are thus higher order transformations for which we do not know more general transformations. We now give three examples found in Ramanujan's second notebook. To repeat by 3.2 each provides a transformation between hypergeometric functions. Theorem 3.. If æ and the multiplier m have degree 5 then =4 =4 æ æ æ æ m = + æ æ æ æ =4 Theorem 3.2. If æ and the multiplier m have degree 7 then =2 =2 =2 =3 m 2 æ æ æ æ æ æ = + 8 æ æ æ æ æ æ Theorem 3.3. If æ and the multiplier m have degree 3 then æ =4 æ =4 æ æ =4 æ æ m = + 4 æ æ æ æ æ æ =6 Theorems 3. and 3.2 can be found in Entries 3xii and 9v respectively in Chapter 9 ë7 pp. 28í282 34ë and Theorem 3.3 coincides with Entry 8iii in Chapter 20 ë7 p. 376ë of Ramanujan's second notebook. See also ë0 p. 72ë. Are these transformations particular to only the function 2 F 2 ; 2 ;;x; or are they special cases of more general transformations? Similar questions can be asked in the alternative theories. Thus in the cubic theory the multiplier m is deæned by m = 2 F 3 ; 2 3 ;;ææ 2F 3 ; 2 3 ;;ææ ; 3.3 where æ has degree n over æ. We cite one example of Ramanujan from his second notebook ë9 p. 24 Thm. 7.5ë. If æ has degree 9 and m is the multiplier deæned by 3.3 for modular equations of degree 9 in the theory of signature 3 then +2æ =3 m =3 + 2 æ =3 We close this section with an open problem about a quasi-transformation formula given by Ramanujan on page 209 of his lost notebook in the pagination of ë5ë. Recall that Kk is deæned by 2.2 and put K 0 = Kk 0 ; where k 0 = p k 2 ; k 0 is called the complementary modulus. If q = e K0 =K and q 0 = e K=K0 then n q 2n+=2 n iq 0 n n 2i Y n=0 + n q 2n+=2 2n+ log q Y n= n iq 0 n

8 8 BRUCE C. BERNDT 2 3 = exp 4 k 3F 2 ; ; ; 3 2 ; 2 ; k2 2F 2 ; 2 ;;k2 3.4 The formulation given in ë5ë is not clear. In particular the deænition of q is faded. Also there is a wavy line to the right of3.4 with the deænition of q on the right of one indentation of the wave and the deænition of q 0 on the right of the other indentation of the wave. The claim 3.4 appears to be a modular-type transformation formula analogous to the transformation formula for theta functions. Thus one naturally thinks of using the Poisson summation formula to eæect a proof but this and other attempts have so far been fruitless. 4. Explicit Evaluations We now turn our attention to explicit evaluations. Observe from 2.5 that an explicit evaluation of one of 'q; Kk; or 2 F 2 ; 2 ;;k2 yields an explicit evaluation of the other two functions. However note that for example if one can determine 'q for a certain speciæed value of q; it may be diæcult to ænd the corresponding value of k Historically there have been several approaches to explicit evaluations but the two most successful have been through the Chowlaí Selberg formula ë56ë and its generalizations and through singular moduli and class invariants; the methods are not unrelated. For evaluations via the former method see for example papers of I. J. Zucker ë7ë and G. S. Joyce and Zucker ë40ë. In our opinion the most successful approach has been through class invariants and singular moduli. This approach also leads to the evaluation of other theta-functions the RogersíRamanujan continued fraction and certain other q-continued fractions. Before discussing class invariants we brieæy mention a recent wonderful result of A. van der Poorten and K. S. Williams ë59ë. Recall the deænition of the Dedekind eta-function ; given by 2.6. Van der Poorten and Williams evaluated b + p d=2a; where d = b 2 4ac and d is a fundamental discriminant ë59 Thm. 9.3ë. They then showed that the ChowlaíSelberg formula and the most far reaching generalization of the ChowlaíSelberg formula found in a paper by J. G. Huard P. Kaplan and Williams ë39ë follow as special cases. Their formulas are quite complicated and are expressed in terms of quantities appearing in the theory of positive deænite binary quadratic forms. Speciæc determinations are therefore diæcult. Now class invariants theta-functions the RogersíRamanujan continued fraction and many other functions in the theory of elliptic functions and modular forms can be expressed in terms of Dedekind eta-functions. The particular products and quotients arising in these contexts usually have more elegant representations without the appearances of parameters from the theory of quadratic forms. Thus it is usually best to directly address the speciæc evaluations of the relevant functions instead of attempting to determine individual values of the relevant eta-functions. To deæne Ramanujan's class invariants or the WeberíRamanujan class invariants ærst set If q =q; q 2 q = exp p n;

9 RAMANUJAN'S GARDEN 9 where n is a positive rational number the two class invariants deæned by G n and g n are G n = 2 =4 q =24 q and g n = 2 =4 q =24 q 4.3 In the notation of Weber ë64ë G n = 2 =4 f p n and g n = 2 =4 f p n It is well known that G n and g n are algebraic; for example see Cox's book ë3 p. 24 Thm. 0.23; p. 257 Thm. 2.7ë. In fact they are frequently units ë9 p. 84 Thm..ë. As above let k = kq; 0 éké; denote the modulus. The singular modulus k n is deæned by k n = ke pn ; where n is a natural number. Following Ramanujan set æ = k 2 and æ n = kn 2 Class invariants and singular moduli are intimately related by the equalities ë9 p. 85 eq..6ë G n = f4æ n æ n g =24 and g n = f4æ n æ n 2 g = Ramanujan calculated a total of 6 class invariants and over 30 singular moduli. See Berndt's book ë9 Chap. 34ë for a description of this work. By 4.4 if one can ænd G n or g n for a particular value of n; then one can ænd æ n by solving a quadratic equation. However generally this simple device yields a rather ugly expression for æ n and so other methods yielding more elegant representations of æ n are desirable. Some simple examples are given by æ 0 = æ 5 = 2 p5 2! 3 p 5+ 2 p p 2 2 ; G 5 = s p5+ 2 A 2 ; p! =4 s 5 ; g0 = 2 + p 5 2 Ramanujan left a huge blooming colorful garden of class invariants singular moduli and explicit evaluations. But it is now time to stroll down some paths where we have not yet been able to espy the æowers. Deæne for positive rational numbers k and n; and fe 2pn=k r k;n = k =4 e p n=kk=2 fe 2 p nk 4.6 fe pn=k s k;n = k =4 e p 4.7 n=kk=24 f k e p nk Our deænition of r k;n is the same as that of J. Yi ë67ë but our deænition of s k;n is slightly diæerent from her deænition of r 0 k;n. Suppose that k =2 Then by 4.6 and 4.4 fe 2pn=2 r 2;n = 2 =4 e p n=2=2 fe 2 p 2n fe p 2n = 2 =4 e p2n=24 fe 2p 2n =2 =4 e p 2n=24 e p 2n ; e 2p 2n = g 2n

10 0 BRUCE C. BERNDT Thus r 2;n is simply equal to the class invariant g 2n Also by 4.7 and 4.4 fe pn s 2;2n = 2 =4 e pn=24 fe 2p n =2 =4 e p n=24 e pn ; e 2pn = G n Hence s 2;2n is simply G n On page 22 in his lost notebook in the pagination of ë5ë Ramanujan deæned n = 3 p f 6 q p 3 qf6 q 3 ; 4.8 where q = e p n=3 Thus n = s 6 3;n On the same page Ramanujan provided a table of eleven recorded values of n ; and ten unrecorded values of n. For example 73 = s + p s 3+ p 73 8 All twenty-one values and several more were established by Berndt Chan S.í Y. Kang and L.íC. Zhang ë5ë. Several methods needed to be devised in order to calculate all the values indicated by Ramanujan. The deænition of n given by 4.8 suggests that the theory and calculation of n are connected with Ramanujan's alternative cubic theory. This indeed is correct and one of the methods used by these authors depends on modular equations in the theory of signature 3. They also showed that the alternative cubic theory and the theory of n lead to new methods for calculating values of the modular j-invariant. Further applications for the values of n have been developed by Chan W.íC. Liaw and V. Tan ë29ë Chan A. Gee and Tan ë26ë and by Berndt and Chan ë3ë. In particular using the theory of n Berndt and Chan ë3ë derived new inænite series for = which are unobtainable by previous methods. One of their series gives a current world record of approximately 73 or 74 digits of per term. In a very brief study of n K. G. Ramanathan ë47ë introduced the arithmetical function n = 3 p f 6 q 2 3 qf 6 q 6 ; 4.9 A 6 where q = e p n=3 Thus by 4.6 n = r 6 3;n It is easy to show ë5 Thm. 3.ë that n and n are related by the equality n n = Gn=3 G 3n 6 In 828 N. H. Abel ëë established the beautiful formula p s p p5+ 2 i 5=2 5+ p 2 2 2i 5 =

11 RAMANUJAN'S GARDEN Note that the left side of 4.0 equals r4;5 2 This example was brought to our attention by F. Hajir and F. Rodriguez Villegas ë37ë who showed that a certain eta-quotient always generates a Kummer extension of the ring class æeld. We have established 4.0 by using basic formulas for and 4 or for fq and fq 4 ë7 p. 24 Entries 2ii ivë along with the evaluation of æ 5 given in 4.5. A systematic study of the functions r 4;n and s 4;n has not been undertaken. A serious study of the case k = 5 should also prove fruitful. In particular such values arise in the explicit evaluation of the RogersíRamanujan continued fraction deæned by Rq = q=5 q q 2 q æææ 4. For example ë7ë ë4 Entry 2.2.7ë the value 5 3=2 s 6 5;7 = e p 7=5 f 6 e p7=5 f 6 e p 35 = p! 3 5 ; 0 is necessary in the evaluation Re p 7=5 = 5 p q p =5 5 ; one of the many explicit values for the RogersíRamanujan continued fraction found in Ramanujan's lost notebook ë5 p. 20ë. For further examples in the case k =5 with applications to the evaluation of the RogersíRamanujan continued fraction see papers by J. Yi ë66ë and K. R. Vasuki and M. S. Mahadeva Naika ë60ë. One of the primary tools for ænding values of G n ;g n n ; and other quotients of eta-functions employs eta-function identities a particular type of modular equation. This has been exploited in particular by Berndt Chan and Zhang ë8ë Berndt Chan Kang and Zhang ë5ë and Yi ë66ë. Ramanujan discovered 23 of these beautiful identities; for proofs see Berndt's book ë8 pp. 204í244ë. For example ë8 p. 22 Entry 62ë if then P = fq q =2 fq 3 and Q = fq 5 q 5=2 fq 5 ; PQ Q 3 P 3 PQ 2 = P Q 4.2 Several others have been found by Yi ë66ë. It is not diæcult to discover such identities by using for example Garvan's etamake package on Maple V. One can always then rigorously prove them by using the theory of modular forms. But these are dull proofs; the proofs in ë66ë and most of them in ë8ë employ Ramanujan's more interesting ideas. Since those working in special functions are most likely more interested in classical theta-functions than in the eta-function especially because of the relations in 2.5 one might ask if it is feasible to calculate directly values of quotients of theta-functions and if theta-analogues of modular equations like 4.2 exist. One can answer such a query in the aærmative but not much isknown at present. In

12 2 BRUCE C. BERNDT his notebooks ë50ë Ramanujan recorded a couple examples like 4.2 with f q replaced by either 'q orq. For example ë8 p. 235 Entry 67ë if then P = 'q 'q 5 and Q = 'q3 'q 5 ; PQ+ 5 3 Q PQ = +3 Q 3 P P P +3P Q Q J. Yi ë68ë is currently systematically searching for other examples. Also recall from 3.2 and 2.5 that the multiplier m can be represented as a quotient of the theta functions '. Representations for m such as those given in Theorems 3.í3.3 can then be used to explicitly determine such quotients. In his notebooks Ramanujan recorded several examples which were ærst proved by Berndt and Chan ë2ë. For example ë2ë ë9 p. 327 Entry 4ë 'e 3 'e = 4p p Since the value 'e = =4 = 3 4 is well-known ë7 p. 03ë 4.3 provides an explicit evaluation of 'e 3 Ramanujan's lost notebook also contains several theorems leading to the explicit evaluation of theta-functions. Most of these theorems have been proved by SooníYi Kang ë4ë. For example if k = RqR 2 q 2 ë5 p. 56ë ë4ë ë4 Chap. ë then ' 2 q 4k k2 ' 2 q 5 = k 2 and 2 q 2 q 5 +k k2 = 4.4 k Thus if we can evaluate Rq and Rq 2 at the requisite values of q; then we can evaluate the quotients of theta-functions above. The introduction of the parameter k is an ingenious device of Ramanujan. It would seem that these ideas can be further exploited. 5. Continued Fractions In Chapters 2 and 6 and the unorganized pages of his second notebook ë50ë ë6 Chap. 2ë ë9 Chap. 32ë Ramanujan recorded many beautiful continued fractions for quotients of gamma functions. For one of the simpler examples we cite Entry 25 of Chapter 2 ë6 p. 40ë. If either n is an odd integer and x is any complex number or if n is any complex number and Re xé0; then 4 x + n +æ 4x n +æ 4 x + n +3æ 4x n +3æ = 4 n 2 2 n n x 2x 2x 2x æææ 5. This particular result is in fact due to Euler ë33 Sec. 67ë but most indeed are original with Ramanujan. For references to other proofs of 5. see ë6 p. 4ë. D. Masson ë43ë ë44ë ë45ë and Zhang ë70ë have shown that Ramanujan's continued fractions for quotients of gamma functions are related to contiguous relations of hypergeometric series. Ramanujan in Chapter 2 and the unorganized pages in

13 RAMANUJAN'S GARDEN 3 his second notebook ë6 pp. 3í63ë ë9 pp. 50í66ë worked out many beautiful limiting cases. For example the continued fraction ë6 p. 55ë 3=+ 3 2 æ æ æ æææ ; was made famous by R.Apery's ë5ë famous proof of the irrationality of3; where s denotes the Riemann zeta-function. Ramanujan was not exhaustive in his search for elegant special cases. We recommend that the particular special and limiting cases be systematically unearthed and brought out into the sunlight. It is natural to ask if q-analogues exist. G. N. Watson ë63ë and D. P. Gupta and Masson ë36ë have established q-analogues of Ramanujan's most general theorem on continued fractions for quotients of gamma functions ë6 p. 63 Entry 40ë. In some cases we know q-continued fractions which share the same features but which apparently are not q-analogues. For example the extremely beautiful Entry 2 in Chapter 6 of the second notebook which wenow state has some of the same salient features as 5.. Suppose that a; b; and q are complex numbers with jabj é and jqj é or that a = b 2m+ for some integer m Then a 2 q 3 ; q 4 b 2 q 3 ; q 4 a 2 q; q 4 b 2 q; q 4 = a bqb aq ab + abq a bq 3 b aq 3 abq æææ 5.2 This is one of my favorite æowers. If jabj é ; then the continued fraction in 5.2 is equivalent to =ab =a q=b=b q=a =a q 3 =b=b q 3 =a =ab + =abq =abq æææ ; which by 5.2 converges to ab q 3 =a 2 ; q 4 q 3 =b 2 ; q 4 q=a 2 ; q 4 q=b 2 ; q 4 Since the continued fraction in 5.2 diverges when jabj =; the hypotheses on a and b in 5.2 cannot be further relaxed. However if jabj é and jqj é ; the continued fraction in 5.2 converges to a 2 =q 3 ;=q 4 b 2 =q 3 ;=q 4 a 2 =q;=q 4 b 2 =q;=q 4 Each beautiful symmetry yields a diæerent view of the æower. Thus what are the q-analogues of Ramanujan's continued fractions for quotients of gamma functions? Also try to ænd continued fractions such as 5.2 which are ëalmost" q-analogues. Several elegant q-continued fractions have representations as q-products. The most famous one of course is the RogersíRamanujan continued fraction deæned

14 4 BRUCE C. BERNDT by 4.. We record several of them. For jqj é ; q 2 ; q 2 q; q 2 = q q 2 + q q 3 q 4 + q 2 q æææ ; 5.3 q 2 ; q 3 q; q 3 = q +q q 3 +q 2 q 5 +q 3 q 7 +q 4 æææ ; 5.4 q; q 2 q 2 ; q 4 2 = q q + q 2 q 3 q 2 + q æææ ; 5.5 q 3 ; q 4 q; q 4 = q q 2 + q 3 q 5 q 4 + q æææ ; 5.6 q 3 ; q 4 q; q 4 = q +q 2 q 3 +q 4 q 5 +q 6 æææ ; 5.7 q; q 5 q 4 ; q 5 q 2 ; q 5 q 3 ; q 5 = q q 2 q æææ ; 5.8 q; q 6 q 5 ; q 6 q 3 ; q 6 2 = q + q 2 q 2 + q 4 q 3 + q æææ ; 5.9 q; q 8 q 7 ; q 8 q 3 ; q 8 q 5 ; q 8 = q + q 2 q 4 q 3 + q æææ 5.0 In examining the bases on the left sides of 5.3í5.0 we ænd that q 7 is missing. Thus we ask if there is a continued fraction representation for q; q 7 q 2 ; q 7 q 4 ; q 7 q 3 ; q 7 q 5 ; q 7 q 6 ; q 7 5. Now D. Bowman and G. Choi ë23ë have recently found a G-continued fraction for q; q 7 q 6 ; q 7 q 3 ; q 7 q 4 ; q 7 Thus maybe one must look for a G-continued fraction for 5.. The product representation for q =5 Rq given in 5.8 was originally discovered by L. J. Rogers ë52ë in 894 and rediscovered by Ramanujan; see his notebooks ë50 Chap. 6 Entry 38iiië ë7 p. 79ë. The function Rq possesses a very beautiful and extensive theory almost all of which was found by Ramanujan. In particular his lost notebook ë5ë contains an enormous amount of material on the Rogersí Ramanujan continued fraction. See papers by Berndt S.íS. Huang J. Sohn and S. H. Son ë9ë and by Kang ë4ë ë42ë for proofs of many of these theorems. In fact the ærst ævechapters of the ærst volume by Andrews and Berndt ë4ë on Ramanujan's lost notebook are devoted to the RogersíRamanujan continued fraction. The continued fraction in 5.0 is called the RamanujaníGordoníGíollnitz continued fraction. It also possesses a beautiful theory much of it developed by Chan and Huang ë27ë. The continued fraction appears in both Ramanujan's second notebook ë50 p. 290ë ë9 p. 50ë and lost notebook ë5 p. 44ë ë4 Cor ë. However the ærst published proof of 5.0 is by A. Selberg ë54 eq. 53ë ë55 pp. 8í9ë. For refereces to further proofs see ë9ë and ë4ë. The continued fraction in 5.9 has also been studied and is called Ramanujan's cubic continued fraction. The ærst appearance of 5.9 is in Chapter 9 of Ramanujan's third notebook ë50ë where several of its properties are given ë7 pp. 345í346 Entry ë. It also appears in his third notebook ë9 p. 45 Entry 8ë. The ærst proof of

15 RAMANUJAN'S GARDEN is by Watson ë62ë and the second is by Selberg ë54 p. 9ë ë55ë. In a fragment published with his lost notebook ë5 p. 366ë Ramanujan writes ëand many results analogous to the previous continued fraction" indicating that there is a theory of the cubic continued fraction similar to that for the RogersíRamanujan continued fraction. Piqued by this remark Chan ë24ë developed an extensive theory of the cubic continued fraction. All the remaining continued fractions cited above can be found in Ramanujan's notebooks or lost notebook. Theories for the continued fractions in 5.3 and 5.5í 5.7 have not been developed but the theories are probably less diæcult than those for the three continued fractions discussed above. The proof of 5.4 is diæcult ë9 pp. 46í47ë and we would expect that if this continued fraction has a theory comparable to those of 5.8í5.0 it will be challenging to discover. Although theories for the RamanujaníGordoníGíollnitz and cubic continued fractions have been developed there are many theorems for the RogersíRamanujan continued fraction for which analogues have not been found for the former two continued fractions and the other continued fractions cited above. We give three examples. In his lost notebook ë5 p. 50ë Ramanujan examined the power series coeæcients v n deæned by X Cq = q =5 Rq = v n q n ; jqj é 5.2 In particular he derived identities for X n=0 n=0 v 5n+j q n ; 0 j These were ærst proved by Andrews ë2ë who also showed that these coeæcients have partition-theoretic interpretations. Let B k;a n denote the number of partitions of n of the form n = b + b 2 + æææ+ b s where b i b i+ ;b i b i+k 2 and at most a of the b i equal. Then for example v 5n = B 37;37 n+b 37;3 n 4 Do the power series coeæcients of any of the other continued fractions cited above have partition-theoretic interpretations? Two of the most important properties for Rq are given by and Rq Rq = fq=5 q =5 fq 5 R 5 q R5 q = f 6 q qf 6 q These equalities were found by Watson in Ramanujan's notebooks ë50ë ë7 pp. 265í 267ë and proved by him ë6ë in order to establish claims about the RogersíRamanujan continued fraction communicated by Ramanujan in his ærst two letters to Hardy ë6ë ë62ë. Some of the continued fractions mentioned above have similar properties. However in his lost notebook ë5 p. 207ë Ramanujan states two-variable generalizations which we oæer in the next theorem.

16 6 BRUCE C. BERNDT Theorem 5.. If and then and P = f0 q 7 ; 5 q 8 +qf 5 q 2 ; 20 q 3 q =5 f 0 q 5 ; 5 q Q = f5 q 4 ; 20 q 3 qfq; 25 q 4 q =5 f 0 q 5 ; 5 q 0 ; 5.7 P Q =+ fq =5 ; q 2=5 q =5 f 0 q 5 ; 5 q P 5 Q 5 =+5PQ+5P 2 Q 2 + fq; 5 q 2 f 5 2 q; 3 q 2 qf 6 0 q 5 ; 5 q 0 ; 5.9 This theorem was ærst proved by Son ë58ë. His task was made more diæcult because Ramanujan did not divulge the identities of P and Q given in 5.6 and 5.7 respectively. Although we forego all details it is not diæcult to show that if we set = above then 5.8 and 5.9 reduce to 5.4 and 5.5 respectively. Very few theorems in the theory of theta functions have two-variable generalizations such as this one. Further investigations along these lines will be diæcult but presumably worthwhile. In his second notebook and especially in his lost notebook Ramanujan recorded very interesting claims about ænite versions of the RogersíRamanujan continued fraction. For example ænite versions at roots of unity are examined and relations with certain class invariants are found. Most of these assertions were proved in a paper by Huang ë38ë. See also a paper by Berndt Huang Sohn and Son ë9ë. Finite versions of the other continued fractions cited abovehave not been examined. On the other side of the coin on page 45 in his lost notebook in the pagination of ë5ë Ramanujan states two very interesting asymptotic formulas for equivalent continued fractions of 5.4 and 5.7 but he apparently has not oæered such a formula for the RogersíRamanujan continued fraction. We state one of the two examples namely for 5.4. Let s denote the Riemann zeta-function and let Ls; denote the Dirichlet n L-function associated with n = 3æ ; where n 3æ denotes the Legendre symbol. For each integer n 2 let Then for xé0 a n = 4nnLn +; 2= p 3 2n+ 3x =3 +e x +e 2x +e 3x æææ = 3æ 2 e Gx ; 3 where as x! 0+; In particular Gx X n= a 2n x 2n a 2 = 08 ; a 4 = 4320 ; a 6 = æ 5.20

17 RAMANUJAN'S GARDEN 7 Both of Ramanujan's asymptotic formulas have recently been proved by Berndt and Sohn ë20ë. They also have established a general theorem containing both examples as special cases. For what continued fractions can one obtain such elegant asymptotic formulas? Can one ænd asymptotic formulas as q = e x approaches other points on the unit circle jqj =? The convergence of 5.3í5.0 on the unit circle is not completely understood. I. Schur ë53ë and Ramanujan ë9 p. 35 Entry 2 Theorem 2.ë have examined the convergence of Rq when q is a primitive root of unity. Zhang ë69ë has examined and 5.0 when q isarootof unity but the other cited continued fractions have yet to be examined. 6. Elliptic Integrals On pages 5í53 in his lost notebook ë5ë Ramanujan recorded several identities involving integrals of eta-functions and incomplete elliptic integrals of the ærst kind. We oæer here one typical example. Recall that fq is deæned in 2.6. Let Then Z q 0 ftft 3 ft 5 ft 5 dt = 5 v = vq =q f 3 qf 3 q 5 f 3 q 3 f 3 q 5 Z 2 tan = p 5 2 tan r p vv 2 5 +vv 2 6. q d' 9 25 sin2 ' 6.2 The reader will immediately realize that these are rather uncommon integrals. S. Raghavan and S. S. Rangachari ë46ë proved all of these formulas but almost all of their proofs use results with which Ramanujan would have been unfamiliar in particular the theory of modular forms which was evidently not known to Ramanujan. For example for four identities including 6.2 Raghavan and Rangachari appealed to diæerential equations satisæed by certain quotients of etafunctions such as 6.. The requisite diæerential equation for 6.2 is given by dv dq = fqfq3 fq 5 fq 5 p 0v 3v 2 +0v 3 + v 4 ; where v is given by 6.. In an eæort to better discern Ramanujan's methods and to better understand the origins of identities like 6.2 Berndt Chan and Huang ë4ë devised proofs independent of the theory of modular forms and other ideas with which Ramanujan would have been unfamiliar. Particularly troublesome were the aforementioned four diæerential equations for quotients of eta-functions for which the authors devised proofs depending on identities for Eisenstein series found in Chapter 2 of Ramanujan's second notebook and eta-function identities such as 4.2. A better understanding and more systematic derivation of these nonlinear diæerential equations likely will play a key role in our understanding of integral identities like 6.2. References ëë N. H. Abel Recherches sur les fonctions elliptiques J. Reine Angew. Math í 90; Oeuvres completes de Niels Henrik Abel Vol. Gríondahl Oslo88 pp. 380í382. ë2ë G. E. Andrews Ramanujan's ëlost" notebook. III. The RogersíRamanujan continued fraction Adv. Math í208. ë3ë G. E. Andrews R. A. Askey and R. Roy Special Functions Cambridge University Press Cambridge999.

18 8 BRUCE C. BERNDT ë4ë G. E. Andrews and B. C. Berndt Ramanujan's Lost Notebook Part I SpringeríVerlag to appear. ë5ë R. Apery Interpolation de fractions continues et irrationalite de certaines constantes Bull. Sect. des Sci. t. III Bibliotheque Nationale Paris98 37í63. ë6ë B. C. Berndt Ramanujan's Notebooks Part II SpringeríVerlag New York989. ë7ë B. C. Berndt Ramanujan's Notebooks Part III SpringeríVerlag New York99. ë8ë B. C. Berndt Ramanujan's Notebooks Part IV SpringeríVerlag New York994. ë9ë B. C. Berndt Ramanujan's Notebooks Part V SpringeríVerlag New York998. ë0ë B. C. Berndt Modular equations in Ramanujan's lost notebook in Number Theory R. P. Bambah V. C. Dumir and R. HansíGill eds. Hindustan Book Co. Delhi999 pp. 55í74. ëë B. C. Berndt S. Bhargava and F. G. Garvan Ramanujan's theories of elliptic functions to alternative bases Trans. Amer. Math. Soc í4244. ë2ë B. C. Berndt and H. H. Chan Ramanujan's explicit values for the classical theta-function Mathematika í294. ë3ë B. C. Berndt and H. H. Chan Eisenstein series and approximations to Illinois J. Math to appear. ë4ë B. C. Berndt H. H. Chan and S.íS. Huang Incomplete elliptic integrals in Ramanujan's lost notebook in qíseries From a Current Perspective M. E. H. Ismail and D. Stanton eds. American Mathematical Society Providence to appear. ë5ë B. C. Berndt H. H. Chan S.íY. Kang and L.íC. Zhang A certain quotient of eta-functions found in Ramanujan's lost notebook Paciæc J. Math. to appear. ë6ë B. C. Berndt H. H. Chan and W.íC. Liaw On Ramanujan's quartic theory of elliptic functions J. Number Theory to appear. ë7ë B. C. Berndt H. H. Chan and L.íC. Zhang Explicit evaluations of the RogersíRamanujan continued fraction J. Reine Angew. Math í59. ë8ë B. C. Berndt H. H. Chan and L.íC. Zhang Ramanujan's class invariants Kronecker's limit formula and modular equations Trans. Amer. Math. Soc í273. ë9ë B. C. Berndt S.íS. Huang J. Sohn and S. H. Son Some theorems on the RogersíRamanujan continued fraction in Ramanujan's lost notebook Trans. Amer. Math. Soc í 277. ë20ë B. C. Berndt and J. Sohn manuscript in preparation. ë2ë J. M. and P. B. Borwein Pi and the AGM Wiley New York987. ë22ë J. M. and P. B. Borwein A cubic counterpart of Jacobi's identity and the AGM Trans. Amer. Math. Soc í70. ë23ë D. Bowman and G. Choi The RogersíRamanujan q-diæerence equations of arbitrary order submitted for publication. ë24ë H. H. Chan On Ramanujan's cubic continued fraction Acta Arith í355. ë25ë H. H. Chan On Ramanujan's cubic transformation formula for 2 F =3; 2=3; ; z Math. Proc. Cambridge Philos. Soc í204. ë26ë H. H. Chan A. Gee and V. Tan Cubic singular moduli the RogersíRamanujan continued fraction and special values of theta functions submitted for publication. ë27ë H. H. Chan and S.íS. Huang On the RamanujaníGordoníGíollnitz continued fraction The Ramanujan J í90. ë28ë H. H. Chan and W.íC. Liaw Cubic modular equations and new Ramanujan-type series for = Paciæc J. Math í238. ë29ë H. H. Chan W.íC. Liaw and V. Tan Ramanujan's class invariant n and a new class of series for = submitted for publication. ë30ë H. H. Chan and Y. L. Ong On Eisenstein series and P m;n= qm2 +mn+2n 2 Proc. Amer. Math. Soc í744. ë3ë D. A. Cox Primes of the Form x 2 + ny 2 Wiley New York989. ë32ë F. J. Dyson A walk through Ramanujan's garden in Ramanujan Revisited G. E. Andrews R. A. Askey B. C. Berndt K. G. Ramanathan and R. A. Rankin eds. Academic Press Boston988 pp. 7í28. ë33ë L. Euler De fractionibus continuis observationes inopera Omnia Ser. I Vol. 4 B. G. Teubner Lipsiae925 pp. 29í349. ë34ë F. G. Garvan Ramanujan's theories of elliptic functions to alternative basesía symbolic excursion J. Symbolic Comput í536.

19 RAMANUJAN'S GARDEN 9 ë35ë E. Goursat Sur l'equation diæerentielle lineaire qui admet pour integrale la serie hypergeometrique Ann. Sci. Ecole Norm. Sup í42. ë36ë D. P. Gupta and D. R. Masson Watson's basic analogue of Ramanujan's Entry 40 and its generalizations SIAM J. Math. Anal í440. ë37ë F. Hajir and F. Rodriguez Villegas Explicit elliptic units I Duke Math. J í 52. ë38ë S.íS. Huang Ramanujan's evaluations of RogersíRamanujan type continued fractions at primitive roots of unity Acta Arith í60. ë39ë J. G. Huard P. Kaplan and K. S. Williams The ChowlaíSelberg formula for genera Acta Arith í30. ë40ë G. S. Joyce and I. J. Zucker Special values of the hypergeometric series Math. Proc. Cambridge Philos. Soc í26. ë4ë S.íY. Kang Some theorems on the RogersíRamanujan continued fraction and associated theta function identities in Ramanujan's lost notebook The Ramanujan J í. ë42ë S.íY. Kang Ramanujan's formulas for the explicit evaluation of the RogersíRamanujan continued fraction and theta-functions Acta Arith í68. ë43ë D. R. Masson Some continued fractions of Ramanujan and MeixneríPollaczek polynomials Canad. Math. Bull í8. ë44ë D. R. Masson Wilson polynomials and some continued fractions of Ramanujan Rocky Mt. J. Math í499. ë45ë D. R. Masson A generalization of Ramanujan's best theorem on continued fractions C. R. Math. Rep. Acad. Sci. Canada í72. ë46ë S. Raghavan and S. S. Rangachari On Ramanujan's elliptic integrals and modular identities in Number Theory Oxford University Press Bombay989 pp. 9í49. ë47ë K. G. Ramanathan On some theorems stated by Ramanujan in Number Theory and Related Topics Tata Institute of Fundamental Research Studies in Mathematics Oxford University Press Bombay989 pp. 5í60. ë48ë S. Ramanujan Modular equations and approximations to Quart. J. Math í372. ë49ë S. Ramanujan Collected Papers Cambridge University Press Cambridge927; reprinted by Chelsea New York962; reprinted by the American Mathematical Society Providence RI ë50ë S. Ramanujan Notebooks 2 volumes Tata Institute of Fundamental Research Bombay 957. ë5ë S. Ramanujan The Lost Notebook and Other Unpublished Papers Narosa New Delhi988. ë52ë L. J. Rogers Second memoir on the expansion of certain inænite products Proc. London Math. Soc í343. ë53ë I. Schur Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbríuche Sitz. Preus. Akad. Wiss. Phys.íMath. Kl í32. ë54ë A. Selberg íuber einige arithmetische Identitíaten Avh. Norske Vid.íAkad. Oslo I. Mat.í Naturv. Kl No í23. ë55ë A. Selberg Collected Papers Vol. I SpringeríVerlag Berlin989. ë56ë A. Selberg and S. Chowla On Epstein's zeta-function J. Reine Angew. Math í0. ë57ë L.íC. Shen On an identity of Ramanujan based on the hypergeometric series 2 F 3 ; 2 3 ; 2 ; xæ J. Number Theory í34. ë58ë S. H. Son Some theta function identities related to the RogersíRamanujan continued fraction Proc. Amer. Math. Soc í2902. ë59ë A. van der Poorten and K. S. Williams Values of the Dedekind eta function at quadratic irrationalities Canad. J. Math í224. ë60ë K. R. Vasuki and M. S. Mahadeva Naika Some evaluations of the RogersíRamanujan continued fractions preprint. ë6ë G. N. Watson Theorems stated by Ramanujan VII Theorems on continued fractions J. London Math. Soc í48. ë62ë G. N. Watson Theorems stated by Ramanujan IX Two continued fractions J. London Math. Soc í237. ë63ë G. N. Watson Ramanujan's continued fraction Proc. Cambridge Philos. Soc í7.

20 20 BRUCE C. BERNDT ë64ë H. Weber Lehrbuch der Algebra Bd. 3 Chelsea New York96. ë65ë E. T. Whittaker and G. N. Watson A Course of Modern Analysis fourth ed. University Press Cambridge966. ë66ë J. Yi Evaluations of the RogersíRamanujan continued fraction Rq by modular equations submitted for publication. ë67ë J. Yi manuscript in preparation. ë68ë J. Yi manuscript in preparation. ë69ë L.íC. Zhang q-diæerence equations and RamanujaníSelberg continued fractions Acta Arith í355. ë70ë L.íC. Zhang Ramanujan's continued fractions for products of gamma functions J. Math. Anal. Applics í52. ë7ë I. J. Zucker The evaluation in terms of -functions of the periods of elliptic curves admitting complex multiplication Math. Proc. Cambridge Philos. Soc í8. Department of Mathematics University of Illinois409 West Green Street Urbana IL 680 USA address berndt@math.uiuc.edu