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

Jongsil Lee, James Mc Laughlin, Jaebum Sohn

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1126-1141
Number of pages16
JournalJournal of Mathematical Analysis and Applications
Volume447
Issue number2
DOIs
Publication statusPublished - 2017 Mar 15

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Continued fraction
Continued Fraction Expansion
Ramanujan
Numerator
Infinite product
Denominator
G-function
Quotient
Converge

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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

In: Journal of Mathematical Analysis and Applications, Vol. 447, No. 2, 15.03.2017, p. 1126-1141.

Research output: Contribution to journalArticle

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