Ramanujan and the regular continued fraction expansion of real numbers

Abstract

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.

Hurwitzian Continued Fractions Containing a Repeated Constant and An Arithmetic Progression

“…(1.6) D. H. Lehmer's result was generalized in recent papers to other Hurwitzian continued fractions [5,7,11]. Some analogous results were found for Tasoev continued fractions [5,6,11,12], defined as continued fractions of the form [a 0 ; a k , . .…”

Some new families of Tasoevian and Hurwitzian continued fractions

“…Komatsu gave several variations of Tasoevian continued fractions in [5], [6], [7] and [8]. In [16], the present author and Nancy Wyshinski derived several variations of Tasoev's continued fraction from known results about q-continued fractions. Two examples of our results from that paper are the following.…”