Ramanujan and the Regular Continued Fraction Expansion of Real Numbers

Abstract

In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m \geq 1$ 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.