In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq,b,λq)/G(a,b,λ) may be derived from each other via Bauer–Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer–Muir transformations converge to the same limit. We also show that these continued fractions may be derived from either Heine's continued fraction for a ratio of ϕ12 functions, or other similar continued fraction expansions of ratios of ϕ12 functions. Further, by employing essentially the same methods, a new continued fraction for G(aq,b,λq)/G(a,b,λ) is derived. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example: (−a,b;q) ∞ −(,− b;q) ∞ (− a b q)∞ + (a, − b; q)∞=(a −b) 1 −a b − (1 − a 2) ( 1 − b 2)q 1 −a b q 2 − (a−b q 2) (b−a q 2)q 1 −a b q 4 − ( 1 − a 2 q 2) ( 1 − b 2 q 2) q 3 1 −a b q 6 − (a−b q 4) (b−a q 4) q 3 1 −a b q 8 − ⋯.
Bibliographical noteFunding Information:
The second author's research was partially supported by a grant from the Simons Foundation (# 209175 to James Mc Laughlin).
The third author's research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2011-0011257 ).
© 2016 Elsevier Inc.
All Science Journal Classification (ASJC) codes
- Applied Mathematics