Classical Constants and Functions: Computations and Continued Fraction Expansions

  • D. V. Chudnovsky
  • G. V. Chudnovsky


This lecture focuses on the arithmetic (diophantine) nature of constants and functions of classical analysis and geometry. We study power series and continued fraction expansion of functions, and related fixed radix and continued fraction expansions of numbers. We try to classify all cases of closed form expressions of continued fraction expansions and present the corresponding identities. At the same time we want to understand what happens when no closed form expression exists.


Riemann Surface Orthogonal Polynomial Linear Differential Equation Elliptic Function Iterate Logarithm 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Borel, Les probabilités, dénombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo, 27, (1909), 247–271.CrossRefzbMATHGoogle Scholar
  2. 2.
    D. V. Chudnovsky and G. V. Chudnovsky, Computational problems in arithmetic of linear differential equations. Some diophantine applications, Lecture Notes Math., Springer, New York, 1383, (1989), 12–49.MathSciNetCrossRefGoogle Scholar
  3. 3.
    N. C. Metropolis, G. Reitwiesner, and J. Von Neumann, Statistical treatment of values of first 2000 decimal places of e and π calculated on the ENIAC, Math Tables Aides Comput., 4, (1950), 109–111.CrossRefGoogle Scholar
  4. 4.
    D. Shanks, J. W. Wrench, Jr., Calculation of π to 100,000 decimals, Math. Comp., v.16 (1962), 76–99.MathSciNetzbMATHGoogle Scholar
  5. 5.
    D. V. Chudnovsky and G. V. Chudnovsky, Algebraic complexities and algebraic curves over finite fields, Proc. Natl. Acad. Sci. USA 84, (1987), 1739–1743.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C. Hermite, Sur la Théorie des Équations Modularies, Oeuvres v.II, Gauthier-Villars, 1910, 38–82.Google Scholar
  7. 7.
    A. Baker, Transcendental Number Theory, Cambridge University Press, 1979.zbMATHGoogle Scholar
  8. 8.
    J. M. Borwein and P. B. Borwein, The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev., 26, (1984) 351–365.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Y. Kanada and Y. Tamura, Calculation of π to 4,196,393 decimals based on Gauss-Legendre algorithm, preprint of Computer Center, University of Tokyo, (1983).Google Scholar
  10. 10.
    D. H. Bailey, The Computation of π to 29,360,000 decimal digits using Borwein’s quartically convergent algorithm, Math. Comput., 50, (1988), 283–296.zbMATHGoogle Scholar
  11. 11.
    C. Caratheodory, Theory of Functions of a complex variable, Chelsea, New York, 1960.Google Scholar
  12. 12.
    S. Ramanujan, Modular equations and approximations to π, Quart. J. Math., 45, (1914), 350–372.Google Scholar
  13. 13.
    W. Gosper, Continued Fraction Arithmetic, preprint MIT AI Lab., (1972).Google Scholar
  14. 14.
    D. V. Chudnovsky and G. V. Chudnovsky, Use of computer algebra for diophantine and differential equations, in “Computer Algebra”, M. Dekker, New York, 1988, 1–82.Google Scholar
  15. 15.
    D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, in “Ramanujan Revisited”, Academic Press, New York, 1988, 375–472.Google Scholar
  16. 16.
    D. V. Chudnovsky and G. V. Chudnovsky, Transcendental methods and theta-functions, in “Proc. Symp. Pure Math.”, American Mathematical Society, Rhode Island, 1989, v. 49, Part II, 167–232.Google Scholar
  17. 17.
    D. V. Chudnovsky and G. V. Chudnovsky, Algebraic complexities and algebraic curves over finite fields, J. Complexity, 4, (1988), 285–316.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D. V. Chudnovsky and G. V. Chudnovsky, Computer Algebra in the Service of Mathematical Physics and Number Theory, in “Computers and Mathematics”, M. Dekker, New York, 1990, 109–232.Google Scholar
  19. 19.
    D. V. Chudnovsky and G. V. Chudnovsky, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, Lecture Notes Math., Springer, New York, 1135, (1985), 52–100.MathSciNetCrossRefGoogle Scholar
  20. 20.
    N. Katz, Nilpotent connections and the monodromy theorem, Publ. Math. IHES, 39, (1970), 355–412.Google Scholar
  21. 21.
    C. L. Siegel, Uder einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math., Kl. 1, (1929).Google Scholar
  22. 22.
    C. L. Siegel, “Transcendental Numbers”, Princeton Univ. Press, Princeton, 1949zbMATHGoogle Scholar
  23. 23.
    D. V. Chudnovsky and G. V. Chudnovsky, Applications of Padé approximations to diophantine inequalities in values of G-functions, Lecture Notes Math., Springer, New York, 1135, (1985), 9–51.MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Weil, “Elliptic Functions According to Eisenstein and Kronecker”, Springer, New York, 1976.CrossRefzbMATHGoogle Scholar
  25. 25.
    D. Knuth, “The Art of Computer Programming”, v. 2 Addison-Wesley, Reading, , 1981.Google Scholar
  26. 26.
    D. V. Chudnovsky and G. V. Chudnovsky, Elliptic formal groups over Z, and F in applications to number theory, computer science and topology, Lecture Notes Math., Springer, New York, 1326, (1988), 11–54.MathSciNetCrossRefGoogle Scholar
  27. 27.
    I. Shiokawa and S. Uchiyama, On some properties of the dyadic Champernowne numbers, Acta. Math. Acad. Sci. Hungar., 26, (1975) 9–27.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    R. Askey, M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoirs Amer. Math. Soc, v. 300, Providence, R.I., 1984.Google Scholar
  29. 29.
    B.C. Brendt, Ramanujan notebooks, Part II, Springer-Verlag, New York, 1989.CrossRefGoogle Scholar
  30. 30.
    G. Szëgo, Orthogonal polynomials, Amer. Math. Soc, Providence, 1975.Google Scholar
  31. 31.
    L. Carlitz, Some orthogonal polynomials related to elliptic functions, Duke Math. J. 27 (1960), 443–460.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    T. Muir, A Treatise on the Theory of Determinants, Dover, New York, 1960.Google Scholar
  33. 33.
    F. Frobenius, L. Stickelberger, Zur Theorie der elliptischen Functionen, F. Frobenius, Gesammelte Abhandlungen, B. I, 335–339, Springer, New York, 1968.Google Scholar
  34. 34.
    F. Frobenius, L. Stickelberger, Uber die Addition und Multiplication der elliptischen Functionen, F. Frobenius, Gesammelte Abhandlungen, B. I, 612–650, Springer, New York, 1968.Google Scholar
  35. 35.
    H.S. Wall, Analytic Theory of Continued Fractions, Chelsea, New York, 1973.Google Scholar
  36. 36.
    D.V. Chudnovsky, Backlund transformations and geometric and complex-analytic background for construction of completely integrable lattice systems, Symmetries in Particle Physics, Plenum Press, 1984, 221–264.Google Scholar
  37. 37.
    D.V. Chudnovsky, G.V. Chudnovsky, Laws of composition of Backlund transformations and the universal form of completely imtegrable systems in dimensions two and three, Proc. Natl. Acad. Sci. U.S.A., 80 (1983), 1774–1777.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    E. Heine, Handbuch der Kugelfunktionen, V. I, Reimer, Berlin, 1878; reprinted Physica-Verlag, Wurzburg, 1961.Google Scholar
  39. 39.
    R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize the classical orthogonal polynomials, Memoirs Amer. Math. Soc, v. 319, Providence, R.I., 1985.Google Scholar
  40. 40.
    V.B. Uvarov, Osvazi polinomov, ortogonalnich s razlichnimi vesami, Dokl. Akad. Nauk SSSR, v. 126 (1959), 33–36 (Russian).MathSciNetzbMATHGoogle Scholar
  41. 41.
    M. Ismail, M. Rahman, The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. (to appear).Google Scholar
  42. 42.
    L. Rogers, On the representation of certain asymptotic series as convergent continued fraction expansions, Proc London Math. Soc. (2) 4 (1907), 72–89. Suppl. Note , p. 395.CrossRefGoogle Scholar
  43. 43.
    T.J. Stieltjes, Recherches sur les fractions continues, Oeuvres complètes, v. II, Groningen, 1918, 402–566.Google Scholar
  44. 44.
    M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math., 70 (1948), 31–74.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    J. Tannery, J. Molk, Eléments de la Théorie des Fonctions Elliptiques, v. I-IV, Paris, 1893–1902, reprinted Chelsea, New York, 1972Google Scholar
  46. 46.
    J. Fay, Theta Functions on Riemann surfaces, Lect. Notes Math., v. 352, 1973.zbMATHGoogle Scholar
  47. 47.
    S. Ramanujan, Notebooks, v. II, Chapter 12, Tata Institute, Bombay, 1957.Google Scholar
  48. 48.
    G.V. Chudnovsky, Rational and Padé approximations to solutions of linear differential equations and monodromy theory, in Proc. Les Houches International Colloq. Complex Analysis and Relativistic Quantum Field Theory, Lecture Notes in Physics, v. 126, Spinger, 1980, 136–169.Google Scholar
  49. 49.
    G.V. Chudnovsky, Ba̋cklund transformations and deformations of linear differential equations with applications to diophantine approximations, Symmetries in Particle Physics, Festschrift in honor of F.Gursey, Plenum Press, NY, 1984, 201–220.Google Scholar
  50. 50.
    G.V. Chudnovsky, Padé’ approximations and the Riemann monodromy problem, Bifurcation Phenomena in Mathematical Physics and Related Topics, D. Reidel, Boston, 1980, 448–540.Google Scholar
  51. 51..
    D.V. Chudnovsky, Riemann monodromy problem, isomonodromy deformation equations and completely integrable systems, 1980, 385–447.Google Scholar
  52. 52.
    D.V. Chudnovsky, Topological and algebraic structures of linear problems associated with completely integrable systems, Lecture Notes in Physics, v.180, Springer, 1983, 65–90.Google Scholar
  53. 53.
    M.L. Mehta, Random Matrices, Academic Press, 1967.zbMATHGoogle Scholar
  54. 54.
    R. Fuchs, Sur quelques equations différentielles linéaires du second order, C.R. Acad. Sci. Paris, v.141 (1907), 555–558.Google Scholar
  55. 55.
    P. Painlevé’, Oeuvres Mathématiques, v.1–3, Paris, 1972.Google Scholar
  56. 56.
    P. Boutroux, Recherche sur les transcendantes de M.Painlevé’ et l’étude asymptotique des équations différentielles du second order, Ann. Ecole Normale, v.30 (1913), 225–375.MathSciNetGoogle Scholar
  57. 57.
    R. Garnier, Sur les singularités irregulières des equations différentielles linéaires J. Math. Pure. Appl., v.2 (1919), 99–200.zbMATHGoogle Scholar
  58. 58.
    R. Nevanlinna, Űber Riemannsche Fla̋chen mit endlich vielen Windungspunkten, Acta Math., v.58 (1932), 295–373.MathSciNetCrossRefGoogle Scholar
  59. 59.
    D. Bessis, A new method in the combinatorics of the topological expansion, Commun. Math. Phys., v.69 (1979), 147–163.MathSciNetCrossRefGoogle Scholar
  60. 60.
    D. J. Gross, A. Migdal, Nonperturbative two dimensional quantum gravity, Phys. Rev. Lett., v.64 (1990), 127–129.MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    D.J. Gross, A. Migdal, A nonperturbative treatment of two dimensional quantum gravity, Princeton preprint PUTP-1159 (1990).Google Scholar
  62. 62.
    R. Dijkgraaf, E. Witten, Mean field theory, topological field theory, and multi-matrix models, Princeton preprint PUTP-1166 (1990).Google Scholar
  63. 63.
    K. Okamoto, Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo, v.33 (1986), 575–618.zbMATHGoogle Scholar
  64. 64.
    H. Flaschka, A. Newell, Monodromy — and spectrum — preserving deformations I, Commun. Math. Phys., v.76 (1980), 65–116.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • D. V. Chudnovsky
    • 1
  • G. V. Chudnovsky
    • 1
  1. 1.Department of MathematicsColumbia UniversityUSA

Personalised recommendations