The Ramanujan Journal https://doi.org/10.1007/s11139-018-0098-4 On the harmonic continued fractions Martin Bunder 1 · Peter Nickolas 1 · Joseph Tonien 2 Received: 27 November 2017 / Accepted: 24 September 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract In this paper, we study the harmonic continued fractions. These form an infinite family of ordinary continued fractions with coefficients t 1 , t 2 , t 3 ,... for all t > 0. We derive explicit formulas for the numerator and the denominator of the convergents. In par- ticular, when t is an even positive integer, we derive the limit value of the harmonic continued fraction. En route, we define and study convolution alternating power sums and prove some identities involving Euler polynomials and Stirling numbers, which are of independent interest. Keywords Continued fractions · Euler polynomials · Stirling numbers · Stirling transform Mathematics Subject Classification 11A55 · 11J70 · 11B73 1 Introduction A continued fraction is an expression of the form a 0 + b 0 a 1 + b 1 a 2 + b 2 . . . . B Joseph Tonien joseph.tonien@uow.edu.au Martin Bunder martin.bunder@uow.edu.au Peter Nickolas peter.nickolas@uow.edu.au 1 School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia 2 School of Computing and Information Technology, University of Wollongong, Wollongong, Australia 123