Asymptotics of Hermite-Padé Polynomials

  • A. I. Aptekarev
  • Herbert Stahl
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)


We review results about the asymptotic behavior (in the strong and weak sense) of Hermite-Padé polynomials of type II (also known as German polynomials). The polynomials appear as numerators and denominators of simultaneous rational approximants. The survey begins with general remarks on Hermite-Padé polynomials and a short summary of the state of the theory in this field.


Riemann Surface Springer Lecture Note Jump Function Markov Function Pade Approximants 
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. [Am]
    Ambraladze, A.U., On convergence of simultaneous Padé approximants, Soobjenya ANGSSR (1985), 119(3), 467–480, (in Russian).Google Scholar
  2. [Ang1]
    Angelesco, A., Sur l’approximation simultanée de plusieurs integrales definies, CR. Paris 167(1918), 629–631.Google Scholar
  3. [Ang2]
    Angelesco, A., Sur deux extensions des fractions continues algébriques, C. R. Paris 168(1919), 262–265.zbMATHGoogle Scholar
  4. [Ang3]
    Angelesco, A., Sur certains polynômes orthogonaux, C. R. Paris 176(1923), 1282–1284.Google Scholar
  5. [Ap1]
    Aptekarev, A.I., Convergence of rational approximations to a set of exponents, Moscow Univ. Math. Bull. 36(1) (1981), 81–86.zbMATHGoogle Scholar
  6. [Ap2]
    Aptekarev, A.I., Padé approximation for the system 1 F 1(l, c, λ i z); l ≤ ik, Moscow Univ. Math. Bull. 36(2) (1981), 73–76.MathSciNetzbMATHGoogle Scholar
  7. [Ap3]
    Aptekarev, A.I., Asymptotic behavior of polynomials of simultaneous orthogonality and of a system of extremal problems for analytic functions, Preprint No. 168, Keldysh Inst. Appl. Math. Acad. Sei. USSR, Moscow (1987), (in Russian).Google Scholar
  8. [Ap4]
    Aptekarev, A.I., Asymptotics of simultaneous orthogonal polynomials in the Angelesco case, Mat. Sb. 136(1), (1988), English transl. in Math. USSR Sb. 64(1) (1989), 57–84.MathSciNetGoogle Scholar
  9. [ApKal1]
    Aptekarev, A.I., Kaljagin, V.A., Analytic properties of two-dimensional P-fraction expansions with periodical coefficients and their simultaneous Padé-Hermite approximants, In: Rational Approximaton and its Application in Mathematics and Physics, Lańcut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer, Lecture Notes 1237, Springer-Verlag, Berlin (1987), 145–160.CrossRefGoogle Scholar
  10. [ApKal2]
    Aptekarev, A.I., Kaljagin, V.A., Analytic behavior of an n-th degree root of polynomials of simultaneous orthogonality, and algebraic functions, Preprint No. 60, Keldysh Inst. Appl. Math. Acad. Sei. USSR, Moscow (1986), (in Russian), MR 88f:41051.Google Scholar
  11. [Ba]
    Baker, A., A note on the Padé table, Proc. Kon. Akad. v. Wet. A’dam Ser. A 69 = Indag. Math. 28(1966), 596–601.Google Scholar
  12. [B-GrM]
    Baker, Jr. G.A., Graves-Morris, P.R., Padé Approximants, Part II: Extension and Applications, Encycl. of Math., and its Applies, vol. 14, Addison-Wesley, London 1981.Google Scholar
  13. [BaLu]
    Baker, Jr. G.A., Lubinsky, D.S., Convergence theorems for rows of differential and algebraic Hermite-Padé approximants, J. Comp, and Appl. Math. 18(1987), 29–52.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Be]
    Beukers, F., Padé approximation in number theory, In: Padé approximation and its applications, Amsterdam 1980, (M.G. de Bruin and H’. van Rossum, Eds.), Springer Lecture Notes 888, Springer-Verlag, Berlin (1981), 90–99.CrossRefGoogle Scholar
  15. [deBr0]
    de Bruin, M.G., Some aspects of simultaneous rational approximation, Banach International Math. Centrum, lecture (1987).Google Scholar
  16. [deBr1]
    de Bruin, M.G., Three new examples of generalized Padé tables which are partly normal, Dept. of Math., Univ. of Amsterdam, report 76–11, Amsterdam (1976).Google Scholar
  17. [deBr2]
    de Bruin, M.G., Some explicit formulas in simultaneous Padé approximation, Lin. Algebra and its Applies. 63(1984), 271–281.CrossRefzbMATHGoogle Scholar
  18. [deBr3]
    de Bruin, M.G., Simultaneous rational approximation to some q-hypergeometric junctions, to appear.Google Scholar
  19. [deBr4]
    de Bruin, M.G., Convergence of some generalized continued fractions, Dept. of Math., Univ. of Amsterdam, report 79–05, Amsterdam (1979).Google Scholar
  20. [deBr5]
    de Bruin, M.G., New convergence results for continued fractions generated by four-term recurrence relations.J. Comp. and Appl. Math. 9(1983), 271–278.CrossRefzbMATHGoogle Scholar
  21. [deBr6]
    de Bruin, M.G., Some convergence results in simultaneous rational approximation to the set of hypergeometric functions 1 F 1(1;c j;z); 1 ≤ jn, In: Padé Approximation and its Applications, Bad Honnef 1983, (H. Werner and H.-J. Bünger, Eds.), Springer Lecture Notes 1071, Springer-Verlag, Berlin, 1984, 12–33.CrossRefGoogle Scholar
  22. [deBr7]
    de Bruin, M.G., Generalized Padé tables and some algorithms therein, In: Proc. of the first French — Polish meeting on Padé approximation and convergence acceleration techniques, Warsaw 1981, (J. Gilewicz, Ed.), CPT-81/PE 1354, CNRS Marseille, 1982.Google Scholar
  23. [deBr8]
    de Bruin, M.G., Simultaneous Padé approximation and orthogonality, In: Polynomes Orthogonaux et Applications, Bar-Le-Duc 1984, (C. Brezinski, A. Draux, A.P. Magnus, P. Maroni and A. Ronveaux, Eds.), Springer Lecture Notes 1171, Springer-Verlag, Berlin, 1985.Google Scholar
  24. [deBr10]
    de Bruin, M.G., Generalized C-fractions and a multidimensional Padé table, Thesis, Univ. of Amsterdam (1974).zbMATHGoogle Scholar
  25. [deBr11]
    de Bruin, M.G., Convergence along steplines in a generalized Padé table, In: Padé and Rational Approximation, (E. B. Saff and R. S. Varga, Eds.), Acad. Press, New York (1977), 15–22.Google Scholar
  26. [deBr12]
    de Bruin, M.G., Convergence of generalized C-fractions, J. Approx. Theory 24(1978), 177–207.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [deBrJa]
    de Bruin, M.G., Jacobsen, L., Modification of generalized continued fractions I: Definition and application to the limit-periodic case, In: Rational Approximation and its Application in Mathematics and Physics, Lańcut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer Lecture Notes 1237, Springer-Verlag, Berlin (1987), 161–176.CrossRefGoogle Scholar
  28. [Chu1]
    Chudnovsky, G.V, Padé approximants to the generalized hyper-geometric functions I, J. de Math. Pures et Applies. 58(1979), 445–476.MathSciNetzbMATHGoogle Scholar
  29. [Chu2]
    Chudnovsky, G.V, Padé approximation and the Riemann monodromy problem. In: Bifurcation phenomena in mathematical physics and related topics, (C. Bardos and D. Bessis, Eds.), Reidel, Dordrecht (1980), 449–510.Google Scholar
  30. [Chu3]
    Chudnovsky, G.V, Rational and Padé approximations to solutions of linear differential equations and the monodromy theory, In: Springer Lecture Notes in Physics 126, Springer-Verlag, Berlin (1980), 136–169.Google Scholar
  31. [Co]
    Coates, J., On the algebraic approximation of functions I-III and IV, Proc. Kon. Akad. v. Wet. A’dam Ser. A 69 and 70 = Indag. Math 28(1966), 421–461, and 29(1967), 205–212.MathSciNetGoogle Scholar
  32. [Gelf]
    Gelfond, A.O., Calculation of limit differences, Moscow, Nauka (1967), (in Russian).Google Scholar
  33. [GoRa1]
    Gonchar, A.A., Rakhmanov, E.A., On convergence of simultaneous Padé approximants for systems of functions of Markov type, Proc. Stekl. Math. Inst. Issue 3(1983), 31–50.Google Scholar
  34. [GoRa2]
    Gonchar, A.A., Rakhmanov, E.A., Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb. 125(1984), 117–127, English transl. in Math. USSR Sb. 53(1986), 119–130.MathSciNetGoogle Scholar
  35. [GoRa3]
    Gonchar, A.A., Rakhmanov, E.A., Equilibrium distributions and the rate of rational approach of analytic functions, Mat. Sb. 134(1987), 306–352, English transl. in Math. USSR Sb. 62(1989), 305–348.Google Scholar
  36. [GrMSa]
    Graves-Morris, P.R., Saff, E.B., Vector-valued rational interpolants, In: Rational Approximation and Interpolation, (P.R. Graves-Morris, E.B. Saff and R.S. Varga, Eds.), Springer Lecture Notes 1105, Springer-Verlag, New York (1984), 227–242.CrossRefGoogle Scholar
  37. [He1]
    Hermite, Ch., Sur la fonction exponentielle, Comptes rendus de l’Acad. Des Sciences t. LXXVII (1873), 18–24, 74–79, 226–233, 285–293 = Oevres t. III (1873), 150–181.Google Scholar
  38. [He2]
    Hermite, Ch., Sur la généralisation des fractions continues algé-briques, Extrait d’une lettre à M. Pincherle, Annali di Matematica 2-ième série, t. XXI (1893), 289–308 = Oevres t. IV (1893), 357–377.Google Scholar
  39. [Hi]
    Hilbert, D., Über die Transzendenz der Zahl e und π, Math Annal. 43(1883), 216–219.MathSciNetCrossRefGoogle Scholar
  40. [IsSa]
    Iserles, A., Saff, E.B., Bi-orthogonality in rational approximation, J. Comp, and Appl. Math 19(1987), 47–54.MathSciNetzbMATHGoogle Scholar
  41. [Ja]
    Jager, H., A multidimensional generalisation of the Padé table, Proc. Kon. Akad. v. Wet. A’dam Ser. A 67 = Indag. Math. 26(1964), 192–249.Google Scholar
  42. [Kal]
    Kalyagin, V.A., On a class of polynomials defined by two orthogonality relations, Mat. Sb. 110(1979), 609–627, English transl. in Math USSR Sb. 38(1981), 563–580.MathSciNetGoogle Scholar
  43. [Kl]
    Klein, F., Elemantarmathematik vom höheren Standpunkt aus, Erster Band, Springer-Verlag, Berlin (1924).Google Scholar
  44. [Lo]
    Lopes, G.L., On the asymptotics of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants, Mat. Sb. 128(1985), 216–229, English transl. in Math. USSR Sb. 56(1987), 207–219.MathSciNetGoogle Scholar
  45. [LoPo]
    Loxton, J.H., van der Poorten, A.J., Multidimensional generalizations of the Padé table, Rocky Mountain J. Math 9(1979), 385–393.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [Mah1]
    Mahler, K., Zur Approximation der Exponentialfunktion aund des Logarithmus I, II, J. Reine Angew. Math. 166(1932), 118–150.CrossRefGoogle Scholar
  47. [Mah2]
    Mahler, K., Applications of some formulas by Hermite to the approximation of exponentials and logarithms, Math. Ann 168(1967), 200–227.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Mah3]
    Mahler, K., Perfect Systems, Compositio Math. 19(1968), 95–166.MathSciNetzbMATHGoogle Scholar
  49. [Ma]
    Mall, J., Grundlagen für eine Theorie der mehrdimensionalen Padéschen Tafel, Inaugural Dissertation, München (1934).Google Scholar
  50. [Ni1]
    Nikishin, E.M., A System of Markov functions, Moscow Univ. Math. Bull. 34(4) (1979), 63–66.MathSciNetzbMATHGoogle Scholar
  51. [Ni2]
    Nikishin, E.M., On simultaneous Padé approximants, Math. USSR Sb. 41(4) (1982), 409–425.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [Ni3]
    Nikishin, E.M., On asymptotics of a linear form for simultaneous Padé approximants, Izvestiya VUZ, Matematika, No. 2 (1986), 33–46, (in Russian).MathSciNetGoogle Scholar
  53. [Ni4]
    Nikishin, E.M., On a set of power series, Siberian Math. J. 22(4) (1981), 164–168 (in Russian).MathSciNetzbMATHGoogle Scholar
  54. [Ni5]
    Nikishin, E.M., On logarithms of natural numbers, Izv. Akad. Nauk USSR, ser. Nat. 43(6) (1979), 1319–1327, English transl. in Math. USSR Izvestiya 15(1980), 523–530.MathSciNetzbMATHGoogle Scholar
  55. [Ni6]
    Nikishin, E.M., On irrationality of values of function F(x, s), Mat. Sb. 109(1979): 410–417, English transl. in Math USSR Sb., 37 (No. 3, 1980), 381–388.MathSciNetGoogle Scholar
  56. [Nut1]
    Nuttall, J., Hermite-Padé approximants to functions meromorphic on a Riemann-surface, J. Approx. Theory, 32(1981), 233–240.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [Nut2]
    Nuttall, J., Asymtotics of diagonal Hermite-Padé approximants, J. Approx. Theory, 42(1984), 299–386.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [Nut3]
    Nuttall, J., Padé polynomial asymptotics from a singular integral equation, Constr. Approx., 6(1990), 157–166.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [NutTr]
    Nuttall, J., Trojan, G.M., Asymtotics of Hermite-Padé approximants for a set of functions with different branch points, Constr. Approx., 3(1987), 13–29.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [Pa1]
    Padé, H., Sur la généralisation des fractions continues algébriques, CR. Hebd. de l’Acad. de France à Paris 118(1894), 848–850.Google Scholar
  61. [Pa2]
    Padé, H., Sur la généralisation des fractions continues algébriques, J. de Math. Pures et Appl. 4-ième série, t. X (1894), 291–329.Google Scholar
  62. [Pa3]
    Padé, H., Mémoire sur les développements en fractions continues de la fonction exponentielle pouvent servir d’introduction à la théorie des fractions continues algébriques, Ann. Sci. Ec. Norm. Sup. (3) 16(1899), 395–426.Google Scholar
  63. [Par1]
    Parusnikov, V.I., The Jacobi-Perron algorithm and simultaneous approximation, Math. USSR Sb. 42(1982), 287–296.CrossRefzbMATHGoogle Scholar
  64. [Par2]
    Parusnikov, V.I., Coefficientwise convergence rate of approximations obtained by Jacobi-Perron algorithm, Siberian Math. J. 25(2) (1985), 935–941.CrossRefGoogle Scholar
  65. [Par3]
    Parusnikov, V.I., Weakly perfect systems and multidimensional continued fractions, Moscow Univ. Math. Bull., 39(2) (1984), 16–21.MathSciNetzbMATHGoogle Scholar
  66. [Par4]
    Parusnikov, V.I., On the convergence of the multidimensional limit-periodic continued fractions, In: Rational approximation and its Application in Mathematics and Physics, Lańcut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer Lecture Notes 1273, Springer-Verlag, Berlin, 1987, 217–227.CrossRefGoogle Scholar
  67. [Pas]
    Paszkowski, S., Hermite-Padé approximation (basic notions and theorems), will appear in J. Comp, and Appl. Math.Google Scholar
  68. [Pi]
    Pineiro Dias, L.R., On simultaneous approximants for some set of Markov functions, Moscow Univ. Math. Bull. No. 2 (1987), 67–70, (in Russian).MathSciNetGoogle Scholar
  69. [So1]
    Sorokin, V.N., Asymptotics of a linear functional form with two logarithms, Uspechi Mat. Nauk USSR, 38(1) (1983), 193–194, (in Russian).MathSciNetzbMATHGoogle Scholar
  70. [So2]
    Sorokin, V.N., Generalization of classic orthogonal polynomials and convergence of simultaneous Padé approximants, Proceeding Petrovsky Seminar (MGU), 11(1986), 125–165, (in Russian).zbMATHGoogle Scholar
  71. [So3]
    Sorokin, V.N., Convergence of simultaneous Padé approximations to functions of Stieltjes type, Izvestiya VUZ, Matematika, No. 7 (1987), 48–56, English transl. in Soviet Math. (Iz VUZ) 31, No 7 (1987), 63–73.MathSciNetGoogle Scholar
  72. [So4]
    Sorokin, V.N., Simultaneous approximation of several linear forms, Moscow Univ. Math. Bull. 38(1983), 53–56.MathSciNetzbMATHGoogle Scholar
  73. [So5]
    Sorokin, V.N., On irrationality of values of hypergeometric functions, Mat. Sb. 127(1985), 245–258, English transl. in Math USSR Sb., 55(No. 1, 1986), 243–257.MathSciNetGoogle Scholar
  74. [St1]
    Stahl, H., Extremal domains associated with an analytic function I and II, Complex. Var. 4(1985), 311–324, 325–338.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [St2]
    Stahl, H., Orthogonal polynomials with complex-valued weight functions I and II, Constr. Approx. 2(1986), 225–240, 241–251.MathSciNetCrossRefzbMATHGoogle Scholar
  76. [St3]
    Stahl, H., Three approaches to a proof of convergence of Padé approximants, In: Rational Approximation and its Applications in Mathematics and Physics, Lańcut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer Lecture Notes 1273, Springer-Verlag, Berlin (1987), 79–124.CrossRefGoogle Scholar
  77. [St4]
    Stahl, H., Asymptotics of Hermite-Padé polynomials and related approximants — A summary of results, In: Non linear Numerical Methods and Rational Approximation, (A. Cuyt, Ed.), Reidel Publ. Corp., Dordrecht (1988), 23–53.Google Scholar
  78. [Wi]
    Widom, H., Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3(1969), 127–232.MathSciNetCrossRefzbMATHGoogle Scholar
  79. [VV1]
    Vleck van, E.B., On the convergence of continued fractions with complex elements, Trans. Amer. Math. Soc. 2(1901), 476–483.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • A. I. Aptekarev
    • 1
  • Herbert Stahl
    • 2
  1. 1.Keldysh InstituteMoscowRussia
  2. 2.TFH/FB2Berlin 65Germany

Personalised recommendations