  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.
