FINITE AND INFINITE ROGERS RAMANUJAN CONTINUED FRACTIONS IN RAMANUJAN S LOST NOTEBOOK

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1 FINITE AND INFINITE ROGERS RAMANUJAN CONTINUED FRACTIONS IN RAMANUJAN S LOST NOTEBOOK BRUCE C. BERNDT, SOON-YI KANG AND JAEBUM SOHN Abstract. Some entries on both finite and infinite Rogers Ramanujan continued fractions in the lost notebook are examined.. Introduction Recall that the Rogers Ramanujan continued fraction is defined by. Rq : q/ + q + q + q 3, q <. + In this paper, we focus on two famous and useful identities for Rq recorded by Ramanujan in his notebooks [9], [, p. 6, Entry iii], namely,. and.3 Rq Rq f q/ q / f q R q R q f 6 q qf 6 q, where.4 f q : q; q : q /4 ητ, q e πiτ, Im τ > 0; the function ητ is the Dedekind eta function. In their examination of the continued fractions in Ramanujan s lost notebook [0], the authors of [] overlooked some entries. The purpose of this paper is to examine these neglected entries on continued fractions and offer a few additional related results. In Section, we establish claims made on page 7 of [0] involving finite Rogers Ramanujan continued fractions, while in Section 3, we turn our attention to theorems on Rq itself. 00 Mathematics Subject Classification: Keywords: Rogers Ramanujan continued fraction

2 BRUCE C. BERNDT, SOON-YI KANG AND JAEBUM SOHN. Finite Rogers Ramanujan Continued Fractions On page 7 of [0], Ramanujan examines four finite Rogers Ramanujan continued fractions that were not covered in []. We state the first as Ramanujan recorded it, although it is perhaps more natural to interchange the hypothesis and conclusion. Entry. p. 7. If x 6 + x 4, then. ix x + ix 3 + x 4 ix x 6 0. Observe that the left-hand side. is simply the sixth partial quotient A 6 /B 6 of /q / Rq, where Rq is the Rogers Ramanujan continued fraction defined in., and where q ix. Ramanujan s claim is that if x 6 + x 4, then A 6 0. Proof. First, for a continued fraction b 0 + a b + a b + + a n b n +, recall the standard recurrence relations for the nth approximant A n /B n [, p. 6] A n b n A n + a n A n, B n b n B n + a n B n, for n, with initial values A, B 0, A 0 b 0, and B 0. We calculate the numerators A n, n 6. To that end, A x ix, A 3 x + x 4 + ix 3 x, A 4 x + x 4 x 6 ix x 3 + x, A x + x 4 x 6 + x 8 ixx 4 + x + x 4, A 6 x x + x 4 + x 6 x 4 ix x + x 4 x 6 x 4. We see that A 6 is the first case when the real and imaginary parts have a common factor, which is x 6 x 4. Ramanujan s Entry. then follows. Do further numerators contain a common factor? We can extend Ramanujan s result with the next theorem. We do not have a general theorem, however. Theorem.. If x 4 + 0, then A 7 /B 7 0, and if x but x ±i, then A 8 /B 8 0. Proof. We note that A 7 x 4 + x x 0 + x 8 x 6 + x 4 x + ixx x + x 4 + x 8 x 6 + x 4 +

3 CONTINUED FRACTIONS IN RAMANUJAN S LOST NOTEBOOK 3 and A 8 x 4 + x 8 x 6 + x 8 x 6 + x 4 x + + ixx 8 x 6 + x 4 x + x 0 x 8 + x 6 x 4. Since x 0 + x + x 8 x 6 + x 4 x +, the result follows. The three remaining entries to be examined here are finite Rogers Ramanujan continued fractions evaluated at roots of unity. All three results are consequences of a table found on page 33 of [], which is incorrectly labelled, for it was erroneously assumed that Ramanujan was employing the same notation on page 46 of [0] as he was on page 7. To remedy this blunder made by the second author of [], we redefine P n x and Q n x, for each positive integer n, by. P n a Q n a : P na, x Q n a, x : + ax + ax + + ax n. Then the following table taken from [, p. 33] is correct. Let ρ denote the least positive residue of n modulo. In each evaluation, x is a primitive nth root of unity. P n P n x P n P n 3 x n, 4 mod x ρn/ x ρn/ 0 n, 3 mod x +ρn/ x +ρn/ 0 n 0 mod 0 0 x n/ + x n/ x n/ + x n/ Entry.3 p. 7. If x is a primitive nth root of unity, with n, 3 mod, then.3 + x x x n Proof. If n, 3 mod, then according to the table above, P n 0, which is precisely the assertion of Entry.3. Entry.4 p. 7. If x is a primitive nth root of unity, with n, 4 mod, then + x + x + + x n.4 Proof. From the table above, if n, 4 mod, then. P n 3 Q n 3 + x + x + + The assertion.4 is equivalent to Q n 0. Now,.6 Q n P n + + x + x + x + + x x n 3 x n 0. x n 3 + x n.

4 4 BRUCE C. BERNDT, SOON-YI KANG AND JAEBUM SOHN By., P n 3 0, and since the approximants of. are identical to those of.6, it follows that Q n 0, which is what we sought to prove. Entry. p. 7. If x is a primitive nth root of unity, then, if n 0 mod,.7 + x + and.8 + x + x + + x + + x n x n Proof. According to the table above, when n 0 mod,.9 P n Q n + x + which is the assertion.7. On the other hand,.0 Q n P n + x + + x + x + + x + + x x n x n 0, x n + x n. Since, by.9, P n 0, and since the partial numerators of.0 are identical to those of.9, we conclude that Q n 0, which is equivalent to the claim Theorems about Rq The first entry that we examine does not appear to be correct as recorded in [0, p. 6]. Appearing in Ramanujan s purported identity is the expression 3. q q 0 0, so that the nth term in this product is x n n. We think that 3. is incorrect, and that Ramanujan mis-recorded the second term of the product representation for f q. On the right-hand side of Ramanujan s formula, we find the function F q, which is not defined by Ramanujan. However, F q is evidently the Rogers Ramanujan continued fraction Rq defined in.. Recall that in his second letter to Hardy, Ramanujan used the same notation, but without the factor q /, to denote the Rogers Ramanujan continued fraction [4, p. 7]. We remark that this entry is difficult to read in [0]. Entry 3. p. 6. Define 3. Qq : q f q f q.

5 CONTINUED FRACTIONS IN RAMANUJAN S LOST NOTEBOOK Then, if F q Rq, Qq / Qq { } + F q F q { { } + F q / F q / } { } + F q F q { { }. + F q / F q / } Proof. For brevity, we set t F q Rq and α /. We employ the identities α t f q 3. t q /0 f q + αq n/ + q, n/ 3.6 t α t q / f q f q n n + αq n + q n, which are found on page 06 in Ramanujan s lost notebook [0], [, pp. ]. Now, with the use of 3., we find that the numerator of the right-hand side of 3.3 is equal to { } { } + F q F q t α t 3.7 q /0 f q f q n. + αq n/ + q n/ In 3.6, replace q by q / and let t Rq / F q /. Then the denominator on the right-hand side of 3.3 equals 3.8 { { + F q / } F q } / t α t f q / q /0 f q n + αq n/ + q n/.

6 6 BRUCE C. BERNDT, SOON-YI KANG AND JAEBUM SOHN If we now divide 3.7 by 3.8, we find that the right-hand side of 3.3 is equal to f q q / f q Qq / qf q f q / Qq, which establishes 3.3. If we use + / in place of α in 3. and 3.6, and proceed as above, we obtain 3.4. In the next entry, Ramanujan offers two values for the Rogers Ramanujan continued fraction Rq. Entry 3. p. 04. Let Then 3.9 and 3.0 where t : Re π and t : Re π. t t + ξ t t + ξ, 3. ξ + ξ ξ. The identity 3.0 was proved by K.G. Ramanathan [6, p. 6]. Proof. Recall the definitions of f q and ητ in.4. In both the proofs of 3.9 and 3.0, we employ the familiar transformation formula for the Dedekind eta function [, p. 43, Entry 7iii] 3. η /τ τ/i ητ. First, from., with the use of 3., we find that t t f e π / e π / fe π ηi /0 ηi / 0 η0i/ ηi / 0 ηi /4 ηi /

7 CONTINUED FRACTIONS IN RAMANUJAN S LOST NOTEBOOK 7 g 0 : where g 0 is Ramanujan s class invariant [3, p. 83], [7], and where g g 0 satisfies the equation [3, p. 0], [, p. 73] g 3 g g +. If we now take ξ /g, we see that 3.3 and 3. are identical. Thus, the proof of 3.9 is complete. Second, using again. and 3., we find that t t g, f e π / e π / fe 0π ηi / ηi ηi / ηi ηi / /4 ηi g 0 g. Since g /ξ, we have completed the proof of 3.0. We now offer some identities related to the identity on page 04 in the lost notebook [0] that we examined above. It follows from 3.9 and 3.0 that 3.4 Re π Re π Re π Re π. We generalize 3.4 and find analogous results. Proposition 3.3. If αβ π with α, β > 0, then 3. Re α Re α Re β Re β Proof. It follows from. that Re α Re α Re β Re β 3.6. f e α/ f e β/ e α/ f e 0α e β/ f e 0β.

8 8 BRUCE C. BERNDT, SOON-YI KANG AND JAEBUM SOHN Note from [, p. 43, Entry 7iii] that if αβ π, then 3.7 e α/ α /4 f e α e β/ β /4 f e β. Applying 3.7 twice, once with α and β/ and once with α/ and β for α and β, respectively, we find that f e α/ f e β/ e α/ f e 0α e β/ f e 0β, which, by 3.6, completes the proof. In fact, 3. is equivalent to the beautiful formula found in Ramanujan s second letter to Hardy [8, p. xxviii], [4, p. 7]: if αβ, with α, β > 0, then { } { } Re πα + Re πβ +. Thus 3.9 and 3.0 may be regarded as refinements of 3.8 when α and β /. Another proof of Proposition 3.3. Let t : tα : Re α and t : t β Re β. After some calculation, we can rewrite 3. as 3.9 tt + t + t + tt t + t t t + t t 4. On the other hand, if we let A + /, we may write 3.8 as 3.0 tt + t + t A. Since A + /A, 3.0 can be also written as 3. tt + t + t + /A. Dividing 3. by tt and adding the result to 3. gives 3. tt + t + t + /tt /t + /t + /At + t /t /t. Using 3.0 in 3., we obtain 3.9. Taking the same procedures for Proposition 3.3, using.3 instead of., we can prove an analogous result for R q. Proposition 3.4. If αβ π / with α, β > 0, then 3.3 R e α R e α R e β R e β.

9 CONTINUED FRACTIONS IN RAMANUJAN S LOST NOTEBOOK 9 Proposition 3.4 is equivalent to + + R e α + + R e β +, where αβ π /, which was established by Ramanathan [6, p. 4, Theorem 3]. Similarly, we establish two formulas invloving Sq : R q. We need to use the following transformation formula for fq instead of 3.7. If αβ π, then [, p. 43, Entry 7iv] 3.4 e α/4 α /4 fe α e β/4 β /4 fe β. Proposition 3.. Let Sq R q. If αβ π with α, β > 0, then 3. Se α Se α + Se β Se β +. Equation 3. is equivalent to Entry 39ii in Chapter 6 of Ramanujan s second notebook [9], [, p. 83, Entry 39ii], namely, + Se α + Se β. Proposition 3.6. If αβ π / with α, β > 0, then 3.6 S e α S e α + S e β S e β +. Equation 3.6 is equivalent to + S e α + S e β, which was also established by Ramanathan [6, p. 4, Theorem 4]. References [] G.E. Andrews and B.C. Berndt, Ramanujan s Lost Notebook, Part I, Springer, New York, 00. [] B.C. Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York, 99. [3] B.C. Berndt, Ramanujan s Notebooks, Part V, Springer-Verlag, New York, 998. [4] B.C. Berndt and R.A. Rankin, Ramanujan: Letters and Commentary, American Mathematical Society, Providence, RI, 99; London Mathematical Society, London, 99. [] L. Lorentzen and H. Waadeland, Continued Fractions, Vol. : Convergence Theory, World Scientific, Paris, 008. [6] K.G. Ramanathan, On Ramanujan s continued fraction, Acta Arith , [7] S. Ramanujan, Modular equations and approximations to π, Quart. J. Math. 4 94, [8] S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 97; reprinted by Chelsea, New York, 96; reprinted by the American Mathematical Society, Providence, RI, 000.

10 0 BRUCE C. BERNDT, SOON-YI KANG AND JAEBUM SOHN [9] S. Ramanujan, Notebooks volumes, Tata Institute of Fundamental Research, Bombay, 97; new edition 0. [0] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 988. [] H. Weber, Lehrbuch der Algebra, dritter Band, Chelsea, New York, 96. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 680, USA address: Department of Mathematics, Kangwon National University, Chuncheon, 00-70, Korea address: Department of Mathematics, Yonsei University, Seoul, 0-749, Korea address: