• Title, Summary, Keyword: continued fraction

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Ramanujan's Continued Fraction, a Generalization and Partitions

  • Srivastava, Bhaskar
    • Kyungpook Mathematical Journal
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    • v.45 no.2
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    • pp.273-280
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    • 2005
  • We generalize a continued fraction of Ramanujan by introducing a free parameter. We give the closed form for the continued fraction. We also consider the finite form giving $n^{th}$ convergent using partition theory.

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SOME REMARKS ON THE PERIODIC CONTINUED FRACTION

  • Lee, Yeo-Rin
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.2
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    • pp.155-159
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    • 2009
  • Using the Binet's formula, we show that the quotient related ratio $l_{1(x)}\;\neq\;0$ for the eventually periodic continued fraction x. Using this ratio, we also show that the derivative of the Minkowski question mark function at the simple periodic continued fraction is infinite or 0. In particular, $l_1({[\bar{1}]})$ = 2 log $\gamma$ where $\gamma$ is the golden mean $(1+\sqrt{5})/2$ and the derivative of the Minkowski question mark function at the simple periodic continued fraction $[\bar{1}]$ is infinite.

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Relations Between Ramanujan's Cubic Continued Fraction and a Continued Fraction of Order 12 and its Evaluations

  • Kumar, Belakavadi Radhakrishna Srivatsa;Vidya, Harekala Chandrashekara
    • Kyungpook Mathematical Journal
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    • v.58 no.2
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    • pp.319-332
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    • 2018
  • In the present paper, we establish relationship between continued fraction U(-q) of order 12 and Ramanujan's cubic continued fraction G(-q) and $G(q^n)$ for n = 1, 2, 3, 5 and 7. Also we evaluate U(q) and U(-q) by using two parameters for Ramanujan's theta-functions and their explicit values.

REPRESENTATIONS OF RAMANUJAN CONTINUED FRACTION IN TERMS OF COMBINATORIAL PARTITION IDENTITIES

  • Chaudhary, Mahendra Pal;Choi, Junesang
    • Honam Mathematical Journal
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    • v.38 no.2
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    • pp.367-373
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    • 2016
  • Adiga and Anitha [1] investigated the Ramanujan's continued fraction (18) to present many interesting identities. Motivated by this work, by using known formulas, we also investigate the Ramanujan's continued fraction (18) to give certain relationships between the Ramanujan's continued fraction and the combinatorial partition identities given by Andrews et al. [3].

General Formulas for Explicit Evaluations of Ramanujan's Cubic Continued Fraction

  • Naika, Megadahalli Sidda Naika Mahadeva;Maheshkumar, Mugur Chinna Swamy;Bairy, Kurady Sushan
    • Kyungpook Mathematical Journal
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    • v.49 no.3
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    • pp.435-450
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    • 2009
  • On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V ($q^3$) and also establish some explicit evaluations using the values of remarkable product of theta-function.

CONTINUED FRACTION AND DIOPHANTINE EQUATION

  • Gadri, Wiem;Mkaouar, Mohamed
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.3
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    • pp.699-709
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    • 2016
  • Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.