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On the harmonic continued fractions

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Abstract

In this paper, we study the harmonic continued fractions. These form an infinite family of ordinary continued fractions with coefficients \(\frac{t}{1}, \frac{t}{2}, \frac{t}{3}, \ldots \) for all \(t>0\). We derive explicit formulas for the numerator and the denominator of the convergents. In particular, 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.

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References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards. U.S. Government Printing Office, Washington, DC (1964)

    MATH  Google Scholar 

  2. Beardon, A.F., Short, I.: The Seidel, Stern, Stolz and Van Vleck theorems on continued fractions. Bull. Lond. Math. Soc. 42(3), 457–466 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bunder, M., Tonien, J.: Closed form expressions for two harmonic continued fractions. Math. Gaz. 101(552), 439–448 (2017)

    Article  MathSciNet  Google Scholar 

  4. Euler, L.: De fractionibus continuis observationes. Comment. Acad. Sci. Imp. Petropolitanae 11, 32–81 (1750)

    Google Scholar 

  5. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  6. Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. Studies in Computational Mathematics. North-Holland Publishing Co., New York (1992)

    MATH  Google Scholar 

  7. Lorentzen, L., Waadeland, H.: Continued Fractions. Convergence Theory, vol. 1. Atlantis Press, London (2008)

    Book  MATH  Google Scholar 

  8. Olds, C.D.: Continued Fractions. The Mathematical Association of America, Providence, RI (1963)

    MATH  Google Scholar 

  9. Seidel, L.: Untersuchungen über die Konvergenz und Divergenz der Kettenbrüche. Habilschrift München, Munich (1846)

    Google Scholar 

  10. Stern, M.A.: Über die Kennzeichen der Konvergenz eines Kettenbruchs. J. Reine Angew. Math. 37, 255–272 (1848)

    Article  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous reviewer for many valuable comments which help improve our paper, and in particularly for pointing out the Euler continued fraction in [6] which we included in the final section.

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Authors and Affiliations

  1. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia

    Martin Bunder & Peter Nickolas

  2. School of Computing and Information Technology, University of Wollongong, Wollongong, Australia

    Joseph Tonien

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  1. Martin Bunder
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  2. Peter Nickolas
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  3. Joseph Tonien
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Correspondence to Joseph Tonien.

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Bunder, M., Nickolas, P. & Tonien, J. On the harmonic continued fractions. Ramanujan J 49, 669–697 (2019). https://doi.org/10.1007/s11139-018-0098-4

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  • DOI: https://doi.org/10.1007/s11139-018-0098-4

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