Continued Fractions with Quadratic Numerators via the Bauer–Muir Transform

Kwang-Wu Chen () and Chia-Hsin Liu
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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

Mathematics, 2025, vol. 13, issue 15, 1-19

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.

Keywords: continued fractions; Bauer–Muir transform; quadratic numerators; Ramanujan Machine; transformation formulas; evaluation identities (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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