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Curious q-Series as Counterexamples in Padé Approximation

  • D. S. Lubinsky
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

Abstract

Basic hypergeometric, or q-series, are usually investigated when |q| < 1. Less common is the case |q| > 1, and the case where q is on the unit circle is extremely rare. It is the latter curious, exotic, choice of q that has yielded a number of interesting examples and counterexamples in Padé approximation, including a counterexample to the Baker-Gammel-Wills Conjecture. We survey some of these, and also pose a number of problems involving q-series for q on the unit circle.

Keywords

Compact Subset Unit Circle Unit Ball Continue Fraction Formal Power Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    G. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.zbMATHGoogle Scholar
  2. [2]
    R. Askey, The q-Gamma and q-Beta Functions, Applicable Analysis, 8 (1978), 125–141.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    G.A. Baker, Essentials of Padé Approximants, Academic Press, New York, 1975.zbMATHGoogle Scholar
  4. [4]
    G.A. Baker, J.L. Gammel, J.G. Wills, An Investigation of the Applicability of the Padé Approximant Method, J. Math. Anal. Appl., 2 (1961), 405–418.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    G.A. Baker and P. R. Graves-Morris, Convergence of Rows of the Padé Table, J. Math. Anal. Appl., 57 (1977), 323–339.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    G.A. Baker and P. R. Graves-Morris, Padé Approximants, 2nd edn., Encyclopaedia of Mathematics and its Applications, Vol. 59, Cambridge University Press, Cambridge, 1996.Google Scholar
  7. [7]
    V. I. Buslaev, Simple Counterexample to the Baker-Gammel- Wills Conjecture, East Journal on Approximations, 4 (2001), 515–517.MathSciNetGoogle Scholar
  8. [8]
    V. I. Buslaev, A.A. Goncar, and S.P. Suetin, On Convergence of Subsequences of the mth Row of a Padé Table, Math. USSR. Sbornik, 48 (1984), 535–540.zbMATHCrossRefGoogle Scholar
  9. [9]
    K.A. Driver, Convergence of Padé Approximants for some q-Hypergeometric Series (Wynn’s Power Series I, II, III ), Thesis, University of the Witwatersrand, 1991.Google Scholar
  10. [10]
    K.A. Driver and D.S. Lubinsky, Convergence of Padé Approximants for a qHypergeometric Series (Wynn’s Power Series I), Aequationes Mathematicae, 42 (1991), 85–106.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    K.A. Driver and D.S. Lubinsky, Convergence of Padé Approximants for a q-Hypergeometric Series (Wynn’s Power Series II), Colloquia Mathematica Societatis Janos Rolyai, Vol. 58, pp. 221–239, Janos Bolyai Math Society, 1990.MathSciNetGoogle Scholar
  12. [12]
    K.A. Driver and D.S. Lubinsky, Convergence of Padé Approximants for a qHypergeometric Series (Wynn’s Power Series III), Aequationes Mathematicae, 45 (1993), 1–23.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    N.J. Fine, Basic Hypergeometric Series and Their Applications, American Mathematical Society Mathematical Surveys and Monographs No. 27, American Mathematical Society, Providence, Rhode Island, 1988.Google Scholar
  14. [14]
    G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.zbMATHGoogle Scholar
  15. [15]
    A.A. Goncar, On Uniform Convergence of Diagonal Padé Approximants, Math. USSR. Sbornik, 46 (1982), 539–559.CrossRefGoogle Scholar
  16. [16]
    G.H. Hardy and J.E. Littlewood, Notes on the Theory of Series (XXIV): A Curious Power Series, Math. Proc. Cambridge Philosophical Soc., 42 (1946), 85–90.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    M.D. Hirschhorn, Partitions and Ramanujan’s Continued Fraction, Duke Math. Journal, 39 (1972), 789–791.MathSciNetzbMATHGoogle Scholar
  18. [18]
    A. Knopfmacher and D.S. Lubinsky, Mathematica Evidence that Ramanujan kills Baker-Gammel- Wills, Applied Mathematics and Computation, 128 (2002), 289–302.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North Holland, 1992.Google Scholar
  20. [20]
    D.S. Lubinsky, Diagonal Padé Approximants and Capacity, J. Math. Anal. Appins., 78 (1980), 58–67.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    D.S. Lubinsky, Will Ramanujan kill Baker-Gammel-Wills? (A Selective Survey of Padé Approximation),(in) Proceedings of IDOMAT98, International Series in Numerical Mathematics, Vol. 132, Birkhäuser, Basel, 1999, pp. 159174.Google Scholar
  22. [22]
    D.S. Lubinsky, Weighted Maximum over Minimum Modulus of Polynomials, with Application to Ray Sequences of Padé Approximants, Constructive Approximation, 18 (2002), 285–308.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    D.S. Lubinsky, Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) Conjecture,to appear in Annals of Mathematics.Google Scholar
  24. [24]
    D.S. Lubinsky and E.B. Saff, Convergence of Padé Approximants of Partial Theta Functions and the Rogers-Szegö Polynomials, Constr. Approx., 3 (1987), 331–361.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    J. Nuttall, Convergence of Padé Approximants of Meromorphic Functions, J. Math. Anal. Appl., 31 (1970), 147–153.MathSciNetCrossRefGoogle Scholar
  26. [26]
    G. Petruska, On the Radius of Convergence of q-Series, Indagationes Mathematicae, 3 (3) (1992), 353–364.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    C. Pommerenke, Padé Approximants and Convergence in Capacity, J. Math. Anal. Appl., 41 (1973), 775–780.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    E.A. Rakhmanov, On the Convergence of Padé Approximants in Classes of Holomorphic Functions, Math. Ussr. Sbornik, 40 (1981), 149–155.zbMATHCrossRefGoogle Scholar
  29. [29]
    A. Sidi, Quantitative and Constructive Aspects of the Generalized Koenig’s and de Montessus’s Theorems for Padé Approximants, J. Comp. Appl. Math., 29 (1990), 257–291.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    H. Stahl, General Convergence Results for Padé Approximants, (in) Approximation Theory VI, (eds. C.K. Chui, L.L. Schumaker, J.D. Ward ), Academic Press, San Diego, 1989, pp. 605–634.Google Scholar
  31. [31]
    H. Stahl, The Convergence of Padé Approximants to Functions with Branch Points, J. Approx. Theory, 91 (1997), 139–204.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    H. Stahl, Conjectures Around the Baker-Gammel-Wills Conjecture: Research Problems 97–2, Constr. Approx., 13 (1997), 287–292.MathSciNetzbMATHGoogle Scholar
  33. [33]
    H. Wallin, The Convergence of Padé Approximants and the Size of the Power Series Coefficients, Applicable Anal., 4 (1974), 235–251.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Proc. Colloquium on Complex Analysis, Joensuu, Finland, 1978, Springer Lecture Notes in Mathematics, Vol. 747, Springer-Verlag, Berlin, 1979, pp. 434–450.Google Scholar
  35. [35]
    P. Wynn, A General System of Orthogonal Polynomials, Quart. J. Math. Oxford Ser., 18 (1967), 81–96.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 2002

Authors and Affiliations

  • D. S. Lubinsky
    • 1
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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