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On the Rogers--Ramanujan Periodic Continued Fraction

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Abstract

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction

$$1 + \frac{qz}{1 + \tfrac{q^2 z}{1 + \cdots}}$$

are studied for q = exp (2 π i τ), where τ is a rational number. It is shown that the function H q to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational τ, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function H q does not exceed one half of its genus.

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REFERENCES

  1. J. Worpitsky, in: Untersuchungen über die Entwicklung der monodromen und monogenen Funktionen durch Kettenbrüche, Friedrichs-Gymnasium rund Realschule Jahresbericht, Berlin, 1865, pp. 3–39.

    Google Scholar 

  2. M. D. Hirschhorn, “Partitions and Ramanujan's continued fraction, ” Duke Math. J., 39 (1972), 789–791.

    Google Scholar 

  3. D. S. Lubinsky, “Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture, ” Ann. of Math., 157 (2003), no. 3, 847–889.

    Google Scholar 

  4. G. A. Baker, J. L. Gammel, and J. G. Wills, “An investigation of the applicability of the Padé approximant method, ” J. Math. Anal. Appl., 2 (1961), 405–418.

    Google Scholar 

  5. H. Stahl, “Conjectures around Baker–Gammel–Wills conjecture: Research problems 97–2, ” Constructive Approx., 13 (1997), 287–292.

    Google Scholar 

  6. H. Stahl, “Diagonal Pad´e approximants to hyperelliptic functions, ” Ann. Fac. Sci. Toulouse. Math., 6 (1996), 121–193.

    Google Scholar 

  7. S. P. Suetin, “On the uniform convergence of diagonal Padé approximants for hyperelliptic functions, ” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 191 (2000), no. 9, 81–114.

    Google Scholar 

  8. S. P. Suetin, “Padé approximations and effective analytic extension of a power series, ” Uspekhi Mat. Nauk [Russian Math. Surveys], 57 (2002), no. 1, 45–142.

    Google Scholar 

  9. V. I. Buslaev, “On the Baker–Gammel–Wills in the theory of Pad´e approximations, ” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 193 (2002), no. 6, 25–38.

    Google Scholar 

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Authors and Affiliations

  1. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Moscow

    V. I. Buslaev

  2. Mathematics Institute, National Academy of Sciences of Ukraine, Ukraine

    S. F. Buslaeva

Authors
  1. V. I. Buslaev
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  2. S. F. Buslaeva
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Buslaev, V.I., Buslaeva, S.F. On the Rogers--Ramanujan Periodic Continued Fraction. Mathematical Notes 74, 783–793 (2003). https://doi.org/10.1023/B:MATN.0000009014.24386.11

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  • DOI: https://doi.org/10.1023/B:MATN.0000009014.24386.11

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