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Continued Fractions with Quadratic Numerators via the Bauer–Muir Transform

Author

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  • Kwang-Wu Chen

    (Department of Mathematics, University of Taipei, Taipei 100234, Taiwan)

  • Chia-Hsin Liu

    (Department of Mathematics, University of Taipei, Taipei 100234, Taiwan)

Abstract

We study a class of continued fraction transformations where the partial numerators are quadratic polynomials and the denominators are linear or constant. Using the Bauer–Muir transform, we establish two theorems that yield structurally distinct but equivalent continued fractions—one with rational coefficients and another with alternating forms. These transformations provide a unified framework for evaluating and simplifying continued fractions, including classical identities such as one of Euler, a recent result by Campbell and Chen, and several conjectures from the Ramanujan Machine involving π and log 2 . We conclude by discussing the potential extension of our methods to more general polynomial cases.

Suggested Citation

  • Kwang-Wu Chen & Chia-Hsin Liu, 2025. "Continued Fractions with Quadratic Numerators via the Bauer–Muir Transform," Mathematics, MDPI, vol. 13(15), pages 1-19, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2332-:d:1707257
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    References listed on IDEAS

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    1. Gal Raayoni & Shahar Gottlieb & Yahel Manor & George Pisha & Yoav Harris & Uri Mendlovic & Doron Haviv & Yaron Hadad & Ido Kaminer, 2021. "Generating conjectures on fundamental constants with the Ramanujan Machine," Nature, Nature, vol. 590(7844), pages 67-73, February.
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