Real Numbers With Polynomial Continued Fraction Expansions

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. This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like $\pi$ and $e$, $\zeta{(k)}$ (for each positive integer $k\geq 2$), various special functions evaluated at integral arguments and various algebraic numbers. We also pose several questions about the nature of the set of real numbers which have a polynomial continued fraction expansion.