Research Article Continued Fractions of Order Six and New Eisenstein Series Identities

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1 Numbers, Article ID 64324, 6 pages Research Article Continued Fractions of Order Six and New Eisenstein Series Identities Chandrashekar Adiga, A Vanitha, and M S Surekha Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore , India Correspondence should be addressed to Chandrashekar Adiga; cadiga@mathsuni-mysoreacin Received 25 February 204; Revised 25 April 204; Accepted 7 May 204; Published 27 May 204 Academic Editor: Ahmed Laghribi Copyright 204 Chandrashekar Adiga et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We prove two identities for Ramanujan s cubic continued fraction and a continued fraction of Ramanujan, which are analogues of Ramanujan s identities for the Rogers-Ramanujan continued fraction We further derive Eisenstein series identities associated with Ramanujan s cubic continued fraction and Ramanujan s continued fraction of order six Introduction Throughout this paper, we assume that q < and for each positive integer n, we use the standard product notation (a; q) 0 :, (a; q) n : (a; q) : j0 ( aq j ) n j0 ( aq j ), n, Srinivasa Ramanujan made some significant contributions to the theory of continued fraction expansions The most beautiful continued fraction expansions can be found in Chapters 2 and 6 of his second notebook [] The celebrated Rogers-Ramanujan continued fraction is defined by [2] () where f (a, b) n a n(n+)/2 b n(n)/2, (ab) n(n)/2 (a n +b n ), ab <, is Ramanujan s general theta function Ramanujan eventually found several generalizations and ramifications of R(q) which can be found in his notebooks [] and lost notebook [3] Recently, Liu [4]andChanetal[5] have established several new identities associated with the Rogers-Ramanujan continued fraction R(q) including Eisenstein series identities involving R(q) The beautiful Ramanujan s cubic continued fraction G(q), first introduced by Srinivasa Ramanujan in his second letter to Hardy [2,pagexxvii],isdefinedby G(q):q /3 f(q,q5 ) f(q 3,q 3 ) (3) R(q):q /5 f(q,q4 ) f(q 2,q 3 ) q /5 q q 2 q3, (2) q /3 q+q 2 q 2 +q 4 q3 +q 6 (4)

2 2 Numbers Adiga et al [6], Bhargava et al [7], Chan [8], and Vasuki et al [9] have proved several elegant theorems for G(q),many of which are analogues of well-known properties satisfied by the Rogers-Ramanujan continued fraction Recently,Vasukietal[0] havestudiedthefollowing continued fraction of order six: X(q):q /4 f(q,q5 ) f(q 2,q 4 ) q /4 ( q 2 ) ( q 3/2 )+ (q/4 q /4 )(q 7/4 q 7/4 ) ( q 3/2 )(q 3 )+ The continued fraction (5) is a special case of a fascinating continued fraction identity recorded by Ramanujan in his second notebook [], [, page 24]Furthermore,they have established modular relations between the continued fractions X(q) and X(q n ) for n 2, 3, 5, 7, and In Section 3 of this paper, we establish two new identities associated with the continued fractions G(q) and X(q), using the quintuple product identity In Section 4, we derive Eisenstein series identities associated with G(q) and X(q) 2 Definitions and Preliminary Results In this section, we present some basic definitions and preliminary results One of the most interesting special cases of f(a, b) is [,Entry22] f (q) : f (q, q 2 ) (5) () n q n(3n)/2 (q; q) (6) n Note that the Dedekind eta function η(τ) q /24 f(q), where q e 2πiτ,Imτ > 0 We need the following three lemmas to prove our main results Lemma (see [, Entry 30, page 46]) One has f (a, b) +f(a, b) 2f(a 3 b, ab 3 ) (7) Lemma 2 (see [,page80]) One has f(b 3 q, q5 B 3 )B2 f( q B 3,B3 q 5 ) f(q 2 ) f(b2,q 2 /B 2 ) f (Bq, q/b) Lemma 3 (see [2, Lemma 2(ii)]) Let m [s/(s r)], l m(sr)r, km(sr)+s,andh mr(m(m)(sr))/2, 0 r<shere[x] denotes the largest integer less than or equal to xthen (i) f(q r,q s )q h f(q l,q k ); (ii) f(q r,q s )() m q h f(q l,q k ) (8) 3 Main Results The Jacobi triple product identity states that n() n q n(n)/2 z n (q; q) (z; q) ( q z ;q), z0 (9) In Ramanujan s notation, the Jacobi triple product identity takes the form f (a, b) (a; ab) (b; ab) (ab; ab) (0) The Jacobi triple product identity was first proved by Gauss [3] Using (3), we have f(e 2iz,qe 2iz )2ie iz () n+ q n(n+)/2 sin (2n + ) z n0 () Putting ae 2iz and bqe 2iz in (0), we obtain f(e 2iz,qe 2iz )(e 2iz )(qe 2iz ;q) (qe 2iz ;q) (q; q) (2) Putting zπ/6and z2π/6,respectively,in(2), we obtain f(e iπ/3,qe iπ/3 ) (e iπ/3 )(qe iπ/3 ;q) (qe iπ/3 ;q) (q; q), (3) f(e i2π/3,qe i2π/3 ) (4) (e i2π/3 )(qe i2π/3 ;q) (qe i2π/3 ;q) (q; q) Multiplying (3) and(4) together and using the identities ( e iπ/3 )(e i2π/3 )i 3, (x) ( xe 2πi/6 )(xe 2πi/6 )(xe 4πi/6 ) (xe 4πi/6 )(xe 6πi/6 )(x 6 ), (5) in the resulting equation and then after some simplifications, we obtain the following identity: f(e iπ/3,qe iπ/3 )f(e i2π/3,qe i2π/3 ) i 3q /4 η 2 (τ) η (6τ) η (2τ) Theorem 4 Let q <, α,andβthen ( + αq n/2 +q n ) ( + βq n/2 +q n ) 4q /24 η 2 (3τ) η 2 (3τ/2) +η 2 (τ/2) η (3τ) G(q), ( + αq n/2 +q n ) ( + βq n/2 +q n ) 4q /24 η (2τ) η (6τ) η 2 (3τ/2) +η 2 (τ/2) η (3τ) X(q) (6) (7) (8)

3 Numbers 3 Proof We may rewrite (3) and(4) as follows: where Then, f(e iπ/3,qe iπ/3 ) (e iπ/3 )q /24 ( + αq n +q 2n ), f(e i2π/3,qe i2π/3 ) (e i2π/3 )q /24 ( + βq n +q 2n ), α2cos π 3, β2cos 2π 3 (αq n +q 2n ) q/24 (βq n +q 2n ) q/24 (9) (20) f(e iπ/3,qe iπ/3 ), (2) ( e iπ/3 ) f(e i2π/3,qe i2π/3 ) (22) ( e i2π/3 ) Multiplying the above two equations together and then using (6), in resulting identity, we find that ( + αq n +q 2n )(βq n +q 2n )q /6 η (6τ) η (2τ) (23) Subtracting (22)from(2), we obtain ( + αq n +q 2n ) q/24 ( + βq n +q 2n ) iπ/3 (f(e,qe iπ/3 ) f(ei2π/3,qe i2π/3 ) ) ( e iπ/3 ) ( e i2π/3 ) (24) Using ()in(24), we deduce that where P (n) ( + αq n +q 2n ) q/2 ( + βq n +q 2n ) () n P (n) q (2n+)2/8, n0 2 sin (2n + )(π/6) ie iπ/6 ( e iπ/3 ) Now, by direct computations, we find that P (6m + 0) 0, P(6m + ) 2, P (6m + 2) 2, (25) 2 sin (2n + )(2π/6) ie i2π/6 ( e i2π/3 ) (26) P(6m + 3) 2, P (6m + 4) 2, P(6m + 5) 0 (27) Therefore, n0 () n P (n) q (2n+)2 /8 2{q (2m+3)2/8 + m0 q (2m+5)2 /8 m0 + q (2m+7)2/8 q (2m+9)2/8 } m0 m0 (28) In the right-hand side of the above equation, changing m to min the first two summations and also changing m to m in the last two summations, we obtain n0 () n P (n) q (2n+)2 /8 2{ m q (2m7)2 /8 m q (2m9)2 /8 } (29) Now, using the definition of f(a, b) in the right-hand side of the above equation, we find that n0 () n P (n) q (2n+)2 /8 2q 9/8 {f (q 9,q 27 )q 2 f(q 3,q 33 )} (30) In the quintuple product identity (8), replacing q by q 6 and then setting Bq,wefindthat f(q 9,q 27 )q 2 f(q 3,q 33 ) f(q2 )f(q 2,q 0 ) (3) f(q 5,q 7 ) Combining (25), (30), and (3), we find that ( + αq n +q 2n ) 2q25/24 ( + βq n +q 2n ) { f(q2 )f(q 2,q 0 ) } f(q 5,q 7 ) (32) Dividing both sides of (32)by (αqn +q 2n )(βq n +q 2n ) andthenusing(23), we obtain ( + αq n +q 2n ) 2q 29/24 ( + βq n +q 2n ) η (2τ) 2 η (6τ) {f(q )f(q 2,q 0 ) } f(q 5,q 7 ) (33) Replacing a by q and b by q 2 in (7), we find that f(q,q 2 )+f(q,q 2 ) 2f (q 5,q 7 ) (34)

4 4 Numbers Now, using the above equation in the right-hand side of (33) andthenchangingq to q /2 throughout, we obtain (7) Equation (8) follows from the following identity: G (q) This completes the proof of Theorem 4 η (2τ) η (6τ) η 2 X (q) (35) (3τ) 4 Eisenstein Series Identities Associated with G(q) and X(q) In this section, we present four Eisenstein series identities associated with G(q) and X(q) Theorem 5 Let q < Then and Lemma 3 in (37), we find that n 2(mod 3) q (q9,q 9,q 8,q 8 ;q 8 ) q (q 5,q 3,q 8,q 8 ;q 8 ) 3 (q 6,q 2,q 3,q 5 ;q 8 ) (q 6,q 2,q 9,q 9 ;q 8 ) +q 5 (q2,q 3,q 8,q 8 ;q 8 ) (q 6,q 2,q 5,q 3 ;q 8 ) (q8 ;q 8 ) 2 {q f(q 9,q 9 ) (q 6,q 2 ;q 8 ) f(q 3,q 5 ) f(q 3,q 5 ) q3 f(q 9,q 9 ) q 2 f(q3,q 5 ) f(q 3,q 5 ) } (39) η3 (8τ) η (6τ) n 2(mod 3) [ G(q 3 ) G(q3 )] (36) Using (4) in(39) and after some simplifications, we obtain (36) Theorem 6 Let q < Then n (mod 6) Proof Changing n to n in the second summation, of the lefthand side of Theorem 5,wehave n 2(mod 3) n n q 3n+ q 8n+6 q n q 3n +q 5n q 6n n q 9n+3 q 8n+6 + n q 5n+5 q 8n+6 (37) Using a corollary of Ramanujan s Ψ summation formula [, Entry 7, page 32] n 5(mod 6) η (2τ) η (8τ) η (36τ) η (6τ) [ X(q 6 ) η 2 (8τ) η (2τ) η (36τ) X(q6 )] Proof Using the identity yq n n0 x n x n y m q mn xq n, x, y <, n0m0 n0 theleft-handsideoftheorem6 canbewrittenas q n q 2n q 4n n (mod 6) y n n n (mod 6) q 2n q 4n +q 5n q 6n (40) (4) n z n aq n (az,q/az,q,q;q), (a, q/a, z, q/z; q) q < z <, (38) n q 6n+ q 36n+6 n q 2n+2 q 36n+6 n q 24n+4 q 36n+6 (42)

5 Numbers 5 Using (38) in(42), we obtain n (mod 6) n 5(mod 6) (q36 ;q 36 ) 2 (q 6,q 30 ;q 36 ) {q (q2,q 24 ;q 36 ) (q 6,q 30 ;q 36 ) q 2 (q8,q 8 ;q 36 ) (q 2,q 24 ;q 36 ) (43) (f(e 2iz,qe 2iz ) f(e 2iz 2,qe 2iz 2 )f(e 2iz 3,qe 2iz 3 )) f(e 2i(z z ) 3,qe 2i(z z ) 3 ) f(e 2i(z z 2 +z ) 3,qe 2i(z z 2 +z ) 3 ) (46) As the proof of this lemma is similar to that of Lemma in [4], we omit the details Using (45), we derive the following Eisenstein series identity Theorem 8 Let q < Then q 4 (q6,q 30 ;q 36 ) (q 2,q 24 ;q 36 ) } Using (5) in(43) and after some simplifications, we obtain (40) Differentiating both sides of (2) andthensettingz0 yield f (, q) 2i(q; q) 3, (44) where f denotes the partial derivative of f with respect to z Now we prove a lemma, which is useful to prove Eisenstein series identities associated with G(q) and X(q) Lemma 7 Consider the following: sin 2nz (e 2iz f (, q 6 )f(q 3,q 3 )f(q,q 5 ) f (e 4iz,q 6 e 4iz )) (4q 2 f(qe 2iz,q 5 e 2iz ) f(q e 2iz,q 7 e 2iz )f(q 2 e 2iz,q 4 e 2iz )) f(q 2 e 2iz,q 8 e 2iz ) (45) Proof For simplicity, we use F(z, q) to denote the logarithmic derivative of iq /8 e iz f(e 2iz,qe 2iz ) with respect to z To prove this lemma we need the following identity, which can be found in [4,Theorem5],[5,Corollary2]: F(z,q)+F(z 2,q)+F(z 3,q)F(z +z 2 +z 3,q) (f (, q) f (e 2i(z z 2 ),qe 2i(z z 2 ) ) f (e 2i(z 2+z 3 ),qe 2i(z 2+z 3 ) )) n( ) η6 (3τ) η 2 (τ) G(q) η4 (3τ) η (6τ) η (2τ) η 2 X(q) (τ) (47) Proof Dividing both sides of (45)byz andthenlettingz 0,weobtain n( ) ((q 6 ;q 6 ) 6 f(q3,q 3 )f(q,q 5 )) (q 2 f(q,q 5 )f(q,q 7 ) f(q 2,q 4 )f(q 2,q 8 )) (48) Using Lemma 3 in (48), we complete the proof of Theorem 8 We use ( /p) to denote the Legendre symbol modulo p Setting z π/3 in (45) and noting that sin (2nπ/3) ( 3/2)(n/3),wefindthat 3 2 ( n 3 ) qn q 2n q 4n +q 5n (e 2iπ/3 f (, q 6 )f(q 3,q 3 ) f (q,q 5 )f(e 4iπ/3,q 6 e 4iπ/3 )) (4q 2 f(e 2i((π/3)+πτ),q 6 e 2i((π/3)+πτ) ) f(e 2i((π/3)πτ),q 6 e 2i((π/3)πτ) )) () (f (e 2i((π/3)+2πτ),q 6 e 2i((π/3)+2πτ) ) f(e 2i((π/3)2πτ),q 6 e 2i((π/3)2πτ) )) (49)

6 6 Numbers Recall the identity [6, Eq (3 )] f(e 2i((π/3)z),qe 2i((π/3)z) ) f(e 2i((π/3)+z),qe 2i((π/3)+z) ) e 2iz (q; q) 3 f(e 6iz,q 3 e 6iz ) e (i2π)/3 (q 3 ;q 3 ) f(e 2iz,qe 2iz ) (50) Using the above identity in (49), we obtain the following Eisenstein series identity Theorem 9 Let q < Then ( n 3 ) qn q 2n q 4n +q 5n η2 (3τ) η (9τ) η (8τ) η 3 G(q) (6τ) Conflict of Interests η (2τ) η (9τ) η (8τ) η 2 X(q) (6τ) (5) The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgments The authors would like to thank the referees for their several helpful comments and suggestions The first author is thankful to the University Grants Commission, Government of India, for the financial support under the Grant F50/2/SAP- DRS/20 The second author is thankful to DST, New Delhi, for awarding INSPIRE Fellowship (no DST/INSPIRE Fellowship/202/22) The third author is thankful to UGC, New Delhi, for UGC-BSR fellowship, under which this work has been done evaluations of theta-functions, Indian Pure and Applied Mathematics,vol35,no9,pp ,2004 [7]SBhargava,KRVasuki,andTGSreeramamurthy, Some evaluations of Ramanujan s cubic continued fraction, Indian Pure and Applied Mathematics,vol35,no8,pp , 2004 [8] H H Chan, On Ramanujan s cubic continued fraction, Acta Arithmetica, vol 73, no 4, pp , 995 [9] K R Vasuki, A A Abdulrawf Kahatan, and G Sharath, On the series expansion of the Ramanujan cubic continued fraction, International Mathematical Combinatorics,vol4,pp 84 95, 20 [0] K R Vasuki, N Bhaskar, and G Sharath, On a continued fraction of order six, Annali dell Universitá di Ferrara, vol 56, no, pp 77 89, 200 [] C Adiga, B C Berndt, S Bhargava, and G N Watson, Chapter 6 of Ramanujan s second notebook: theta functions and q- series, Memoirs of the American Mathematical Society,vol35, pp 9, 985 [2] C Adiga and N S A Bulkhali, Identities for certain products of theta functions with applications to modular relations, Journal of Analysis & Number Theory,vol2,no,pp 5,204 [3]CFGauss, Hundert Theorems über die neuen Transscendenten, in Werke, 3, pp , Königliche Gesellschaft der Wissenschaften zu Göttingen, Göttingen, Germany, 876 [4] Z-G Liu, A three-term theta function identity and its applications, Advances in Mathematics,vol95,no,pp 23,2005 [5] S McCullough and L-C Shen, On the Szegő kernelofan annulus, Proceedings of the American Mathematical Society,vol 2, no 4, pp 2, 994 [6] Z-G Liu, A theta function identity and its implications, Transactions of the American Mathematical Society,vol357,no 2, pp , 2005 References [] S Ramanujan, Notebooks Vols, 2,TataInstituteofFundamental Research, Bombay, India, 957 [2] S Ramanujan, Collected Papers, Cambridge University Press, Cambridge, UK, 927, repriented by Chelsea, New York, NY, USA, 962; reprinted by the American Mathematical Society, Providence, RI, USA, 2000 [3] S Ramanujan, The Lost Notebook and Other Unpublished Papers, Springer, Berlin, Germany; Narosa Publishing House, New Delhi, India, 988 [4] Z-G Liu, A theta function identity and the Eisenstein series on Γ 0 (5), the Ramanujan Mathematical Society, vol22, no 3, pp , 2007 [5] H H Chan, S H Chan, and Z-G Liu, The Rogers-Ramanujan continued fraction and a new Eisenstein series identity, Journal of Number Theory,vol29,no7,pp ,2009 [6] C Adiga, T Kim, M S Mahadeva Naika, and H S Madhusudhan, On Ramanujan s cubic continued fraction and explicit

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