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 π and e, ζ(k) (for each positive integer k ≥ 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.

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors.By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods.By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers-Ramanujan-type identities.

Abstract. In some recent papers, the authors considered regular continued fractions of the formwhere a 0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases m = 1 and m = 2 and also use known results about other q-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions.