The continued fractions found in the unorganized portions of Ramanujan’s notebooks

Andrews, George E.; Berndt, Bruce C.; Jacobsen, Lisa; Lamphere, Robert L.

doi:10.1090/memo/0477

Cited by 45 publications

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“…) ∞ (e −x ; e −3x ) ∞ , by (1.2), which can be found in Ramanujan's second notebook [20] and which was first proved by Andrews, Berndt, Jacobsen, and Lamphere [5], [9, p. 46]. This expression is the case k = 3 in Theorem 2.2, since M 1 (χ) = −1.…”

Section: Where G(χ) Is the Gaussian Sum Defined Bymentioning

confidence: 59%

Asymptotic Formulas for Two Continued Fractions in Ramanujan's Lost Notebook

Berndt¹,

Sohn

2002

J. London Math. Soc.

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Section: Where G(χ) Is the Gaussian Sum Defined Bymentioning

confidence: 59%

Asymptotic Formulas for Two Continued Fractions in Ramanujan's Lost Notebook

Berndt¹,

Sohn

2002

J. London Math. Soc.

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“…This was stated by Ramanujan without proof, and was proved by Andrews, Berndt, Jacobsen and Lamphere in 1992 ( [1]). This leaves the question of convergence on the unit circle.…”

Section: Theorem 1 (Worpitzky) Let the Continued Fraction Kmentioning

confidence: 70%

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Bowman

McLaughlin

2003

Trans. Amer. Math. Soc.

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Abstract. This paper studies ordinary and general convergence of the RogersRamanujan continued fraction.Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, e 1 (t), e 2 (t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted bywhere φ = (It is shown that if y ∈ Y S , then the Rogers-Ramanujan continued fraction R(y) diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points G ⊂ Y S such that if y ∈ G, then R(y) does not converge generally.It is further shown that R(y) does not converge generally for |y| > 1. However we show that R(y) does converge generally if y is a primitive 5m-th root of unity, for some m ∈ N. Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

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“…For example, let f (q) = f B 5 (q). For q = e 2πiτ with Im (τ ) > 0, one can show the following (see Theorems 7.5.1, 7.5.3, and 7.5.4 in [5] and Entry 24 and Entry 25 in [3]): …”

Section: Proofmentioning

confidence: 95%

Birth-death processes and $q$-continued fractions

Feng

Kirsch²,

Villella

et al. 2012

Trans. Amer. Math. Soc.

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Abstract. In the 1997 paper of Parthasarathy, Lenin, Schoutens, and Van Assche, the authors study a birth-death process related to the Rogers-Ramanujan continued fraction r(q). We generalize their results to establish a correspondence between birth-death processes and a larger family of q-continued fractions. It turns out that many of these continued fractions, including r(q), play important roles in number theory, specifically in the theory of modular forms and q-series. We draw upon the number-theoretic properties of modular forms to give identities between the transition probabilities of different birth-death processes.

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The continued fractions found in the unorganized portions of Ramanujan’s notebooks

Cited by 45 publications

References 0 publications

Asymptotic Formulas for Two Continued Fractions in Ramanujan's Lost Notebook

Asymptotic Formulas for Two Continued Fractions in Ramanujan's Lost Notebook

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Birth-death processes and $q$-continued fractions

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