James Mc Laughlin
580 California St., Suite 400
San Francisco, CA, 94104
REAL NUMBERS WITH POLYNOMIAL CONTINUED FRACTION EXPANSIONS
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Abstract
In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit.
FAQs
AI
What roles do polynomial continued fractions play in understanding famous constants?
The research indicates that polynomial continued fractions can express constants like π with predictable patterns, contrasting their traditional unpredictable representations. Specifically, Brouncker's continued fraction for π serves as an early example of this formulation.
How does Tietze's Criterion classify irrational limits of continued fractions?
The study shows that Tietze's Criterion applies when sequences of integers and positive integers converge, establishing irrationality for continued fractions under specific conditions from N0. This builds on classical results by Legendre, extending their applications to polynomial sequences.
What synthetic way does Pincherle's theorem offer for constructing polynomial continued fractions?
Using Pincherle's theorem, the research demonstrates that one can create infinite families of polynomial continued fractions converging to the same limit, relieving the necessity for polynomial coefficients. This provides a fresh perspective on convergence claims previously stated for individual fractions.
How is Euler's transformation generalized in polynomial continued fraction contexts?
The paper generalizes Euler's transformation by asserting that converging series can be expressed as polynomial continued fractions, shown via specific structural examples for functions representing classic series like π/4. This approach enhances the link between series and continued fractions through polynomial terms.
What does the study reveal about the classification of polynomial continued fractions?
The findings suggest that understanding polynomial continued fractions could lead to a classification system for converging fractions, aiming for a comprehensive base set with convergences documented in hypergeometric series. This emphasizes the potential for higher polynomial degrees in constructing such fractions.
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