### Abstract

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 − ⋯.

Original language | English |
---|---|

Pages (from-to) | 1126-1141 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 447 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 Mar 15 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*447*(2), 1126-1141. https://doi.org/10.1016/j.jmaa.2016.10.052

}

*Journal of Mathematical Analysis and Applications*, vol. 447, no. 2, pp. 1126-1141. https://doi.org/10.1016/j.jmaa.2016.10.052

**Applications of the Heine and Bauer–Muir transformations to Rogers–Ramanujan type continued fractions.** / Lee, Jongsil; Mc Laughlin, James; Sohn, Jaebum.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Applications of the Heine and Bauer–Muir transformations to Rogers–Ramanujan type continued fractions

AU - Lee, Jongsil

AU - Mc Laughlin, James

AU - Sohn, Jaebum

PY - 2017/3/15

Y1 - 2017/3/15

N2 - 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 − ⋯.

AB - 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 − ⋯.

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U2 - 10.1016/j.jmaa.2016.10.052

DO - 10.1016/j.jmaa.2016.10.052

M3 - Article

VL - 447

SP - 1126

EP - 1141

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -