International Journal of Advanced Research in Mathematics and Applications Volume: 1 Issue: 1 08-Jan-2014,ISSN_NO: xxxx-xxx An Interesting q-Continued Fractions of Ramanujan S. N. Fathima 1 , T. Kathiravan and Yudhisthira Jamudulia Department of Mathematics,Ramanujan School of Mathematics, Pondicherry University fathima.mat@pondiuni.edu.in,kkathiravan98@ail.com,yudhis4581@yahoo.co.in. Abstract— In this paper, we establish an interesting q-identity and an integral representation of a q-continued fraction of Ramanujan. We also compute explicit evaluation of this continued fraction and derive its relation with Ramanujan G¨ollnitz -Gordon continued fraction. 2000 Mathematics Subject Classification: 11A55. Keywords: Continued fractions, Modular Equations. 1 Introduction Ramanujan a pioneer in the theory of continued fraction has recorded several in the process rediscovered few continued fractions found earlier by Gauss, Eisenstein and Rogers in his notebook [10]. In fact Chapter 12 and Chapter 16 of his Second Notebook [10] is devoted to continued fractions. Proofs of these continued fractions over years are given by several mathematician, we mention here specially G.E. Andrews[3], C. Adiga, S. Bhargava and G.N. Watson [1] whose works have been compiled in [4] and [5]. The celebrated Roger Ramanujan continued fraction is defined by 1 qq 2 q 3 R(q) := 1 + 1 + 1 + 1 + · · · ,|q| < 1. On page 365 of his lost notebook [11], Ramanujan recorded five modular equations relating R(q) with R( −q), R(q 2 ), R(q 3 ) R(q 4 ) and R(q 5 ). The well known Ramanujan’s cubic continued fraction defined by q 1/3 q + q 2 q 2 + q 4 q 3 + q 6 J(q) := + · · · ,|q| < 1. 1 + 1 + 1 + 1 1 Supported by UGC Grant No. F41-1392/2012/(SR) is recorded on page 366 of his lost notebook [11]. Several new modular equation relating J(q) with J(−q), J(q 2 ) and J(q 3 ) are established by H.H. Chan [8]. Similarly the Ramanujan G¨ollnitz-Gordon continued fraction K(q) defined by q1/2 q 2 q 4 q 6 K(q) := + + + + · · · ,|q| < 1, 1 + q 1 + q 3 1 + q 5 1 + q 7 satisfies several beautiful modular relations. One may see traces of modular equation related to K(q) on page 229 of Ramanujan’s lost notebook [11]. Further works related to K(q) in recent years have been done by various authors including Chan and S.S Huang [9] and K.R. Vasuki and B.R. Srivatsa Kumar [12]. Motivated by these works in this paper we study the Ramanujan continued fraction M(q) := q1/2 q(1 − q) q(1 − q 3 ) 2 q(1 − q 5 ) 2 , q | < 1 1 q + 1 + q 2 + (1 − − q)(1 + q 4 ) + (1 − q)(1 + q 6 ) + · · · | (q 4 ; q 4 ) 2 ISRJournals and Publications Page 63