On Some Continued Fraction Identities of Srinivasa Ramanujan

Abstract

Abstract. The main purpose of this note is to state and prove, in a simple, unified manner, several 17-continued fraction expansions found in Ramanujan's "lost" notebook. This is related to some recent works of G. E. Andrews and M. D. Hirschhorn.

On some continued fraction expansions of the Rogers–Ramanujan type

“…Later, Bhargava and Adiga proved this identity in [7]. In what follows, applying our method, we can give an elementary proof of this continued fraction identity (3.1).…”

“…s 2k = q 8k 2 +4k n≥0 z n q n 2 +2n(2k+1) (−q 8 ; q 8 ) k−1 (q 2 ; q 2 ) n (−q 2 ; q 2 ) n+4k−1 (−q 4 ; q 8 ) k , k ≥ 1, s 2k+1 = q 8k 2 +12k+4 n≥0 z n q n 2 +4n(k+1) (−q 4 ; q 8 ) k (q 2 ; q 2 ) n (−q 2 ; q 2 ) n+4k+1 (−q 8 ; q 8 ) k , z 2 G(z) F (z) = z 2 1 + zq + z 2 q 4 (1 + q 4 ) (1 + q 2 )(1 + q 4 ) − zq 7 + z 2 q 12 (1 + q 8 ) (1 + q 4 )(1 + q 6 )(1 + q 8 ) + zq 5 (1 − q 12 ) + z 2 q 20 (1 + q 4 )(1 + q 12 ) (1 + q 8 )(1 + q 10 )(1 + q 12 ) + zq 7 (1 − q 20 ) + z 2 q 28 (1 + q 8 )(1 + q 16 ) (1 + q 12 )(1 + q 14 )(1 + q 16 ) + zq 9 (1 − q 28 ) + · · · + z 2 q 8n+4 (1 + q 4n−4 )(1 + q 4n+4 ) (1 + q 4n )(1 + q 4n+2 )(1 + q 4n+4 ) + zq 2n+3 (1 − q 8n+4 ) + · · · . Theorem 4.5 Let…”

On the values of continued fractions: q-series

“…appears in Ramanujan's lost notebook and is studied in [9] formula ðIV Þ R : The value of (28) may deduced from (18) and (20) and we note that result (4) comes as a special case.…”

“…This should be compared to the earlier considerations [1,2,5,[7][8][9][10][11][12][13][14][15][16]19,20], where frequently the polynomials a n ðq; tÞ and b n ðq; tÞ have quite low degrees with respect to t: In [6] a value of continued fraction with degrees 4 and 2 for a n ðq; tÞ and b n ðq; tÞ; respectively, is given by a quotient of q-hypergeometric series.…”

On Bilateral Basic Hypergeometric Series and Continued Fractions

“…Chapters 1, 2 and 3 of Agarwal [6] deals with a number of interesting results of Ramanujan's on continued fractions. Most of the continued fractions for the quotient of two basic hypergeometric series namely 2 φ 1 extended for the quotient of two basic bilateral series of 2 ψ 2 type and a number of author's namely Denis [9,10], Singh [12,13,14], Agarwal [7,8], Bhargava and Adiga [1], Srivastava [2,3], Srivastava and Mishra [4], Srivastava et al [5] and many more established important application for basic bilateral series and continued fractions. In the present paper we have made an attempt to establish certain results involving bilateral basic hypergeometric series and continued fractions.…”