×

zbMATH — the first resource for mathematics

Convergence analysis of the summation of the factorially divergent Euler series by Padé approximants and the delta transformation. (English) Zbl 1325.65008
Summary: Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series \(\mathcal{E}(z) \sim \sum_{n = 0}^\infty(- 1)^n n! z^n\) is a very important model for the ubiquitous factorially divergent perturbation expansions in theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Padé approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results.

MSC:
65B10 Numerical summation of series
41A21 Padé approximation
40D05 General theorems on summability
40G99 Special methods of summability
Software:
DLMF
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allen, G. D.; Chui, C. K.; Madych, W. R.; Narcowich, F. J.; Smith, P., Padé approximation of Stieltjes series, J. Approx. Theory, 14, 302-316, (1975) · Zbl 0323.30044
[2] Baker, G. A., Essentials of Padé approximants, (1975), Academic Press New York · Zbl 0315.41014
[3] Baker, G. A.; Graves-Morris, P., Padé approximants, (1996), Cambridge U. P. Cambridge · Zbl 0923.41001
[4] Barbeau, E. J., Euler subdues a very obstreperous series, Am. Math. Mon., 86, 356-372, (1979) · Zbl 0422.40004
[5] Barbeau, E. J.; Leah, P. J., Euler’s 1760 paper on divergent series, Hist. Math., 3, 141-160, (1976) · Zbl 0325.01011
[6] Beals, R.; Wong, R., Special functions: A graduate text, (2010), Cambridge U. P. Cambridge · Zbl 1222.33001
[7] Bender, C. M.; Orszag, S. A., Advanced mathematical methods for scientists and engineers, (1978), McGraw-Hill New York · Zbl 0417.34001
[8] Bender, C. M.; Weniger, E. J., Numerical evidence that the perturbation expansion for a non-Hermitian \(\mathcal{PT}\)-symmetric Hamiltonian is Stieltjes, J. Math. Phys., 42, 2167-2183, (2001) · Zbl 1014.81018
[9] Bender, C. M.; Wu, T. T., Anharmonic oscillator, Phys. Rev., 184, 1231-1260, (1969)
[10] Bender, C. M.; Wu, T. T., Large-order behavior of perturbation theory, Phys. Rev. Lett., 27, 461-465, (1971)
[11] Bender, C. M.; Wu, T. T., Anharmonic oscillator. II. A study in perturbation theory in large order, Phys. Rev. D, 7, 1620-1636, (1973)
[12] Borel, E., Mémoires sur LES séries divergentes, Ann. Sci. Éc. Norm. Super., 16, 9-136, (1899) · JFM 30.0230.03
[13] Borel, E.; Critchfield, C. L.; Vakar, A., Lectures on divergent series, (1975), Los Alamos Scientific Laboratory Los Alamos, Translation LA-6140-TR
[14] Borghi, R., Evaluation of diffraction catastrophes by using weniger transformation, Opt. Lett., 32, 226-228, (2007)
[15] Borghi, R., Joint use of the weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals, Phys. Rev. E, 78, 026703, (2008), (11 pp.)
[16] Borghi, R., On the numerical evaluation of cuspoid diffraction catastrophes, J. Opt. Soc. Am. A, 25, 1682-1690, (2008)
[17] Borghi, R., Summing Pauli asymptotic series to solve the wedge problem, J. Opt. Soc. Am. A, 25, 211-218, (2008)
[18] Borghi, R., Joint use of the weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. higher-order transformations, Phys. Rev. E, 80, 016704, (2009), (15 pp.)
[19] Borghi, R., Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials, Appl. Numer. Math., 60, 1242-1250, (2010) · Zbl 1203.30006
[20] Borghi, R., On the numerical evaluation of umbilic diffraction catastrophes, J. Opt. Soc. Am. A, 27, 1661-1670, (2010)
[21] Borghi, R., Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four, J. Opt. Soc. Am. A, 28, 887-896, (2011)
[22] Borghi, R., Optimizing diffraction catastrophe evaluation, Opt. Lett., 36, 4413-4415, (2011)
[23] Borghi, R., Numerical computation of diffraction catastrophes with codimension eight, Phys. Rev. E, 85, 046704, (2012), (14 pp.)
[24] Borghi, R.; Gori, F.; Guattari, G.; Santarsiero, M., Decoding divergent series in nonparaxial optics, Opt. Lett., 36, 963-965, (2011)
[25] Borghi, R.; Santarsiero, M., Summing Lax series for nonparaxial beam propagation, Opt. Lett., 28, 774-776, (2003)
[26] Bornemann, F.; Laurie, D.; Wagon, S.; Waldvogel, J., The SIAM 100-digit challenge: A study in high-accuracy numerical computing, (2004), Society of Industrial Applied Mathematics Philadelphia · Zbl 1060.65002
[27] Brezinski, C., Accélération de la convergence en analyse numérique, (1977), Springer-Verlag Berlin · Zbl 0352.65003
[28] Brezinski, C., Algorithmes d’accélération de la convergence - étude numérique, (1978), Éditions Technip Paris · Zbl 0396.65001
[29] Brezinski, C., Rational approximation to formal power series, J. Approx. Theory, 25, 295-317, (1979) · Zbl 0401.41013
[30] Brezinski, C., Padé-type approximation and general orthogonal polynomials, (1980), Birkhäuser Basel · Zbl 0418.41012
[31] Brezinski, C., History of continued fractions and Padé approximants, (1991), Springer-Verlag Berlin · Zbl 0714.01001
[32] Brezinski, C., Extrapolation algorithms and Padé approximations: a historical survey, Appl. Numer. Math., 20, 299-318, (1996) · Zbl 0854.65001
[33] Brezinski, C.; Redivo Zaglia, M., Extrapolation methods, (1991), North-Holland Amsterdam · Zbl 0814.65001
[34] Brezinski, C.; Redivo Zaglia, M., Extensions of Drummond’s process for convergence acceleration, Appl. Numer. Math., 60, 1231-1241, (2010) · Zbl 1201.65005
[35] Brezinski, C.; Redivo Zaglia, M.; Weniger, E. J., Special issue: approximation and extrapolation of convergent and divergent sequences and series (CIRM, luminy, France, 2009), Appl. Numer. Math., 60, 1183-1464, (2010) · Zbl 1205.65005
[36] Bromwich, T. J.I., An introduction to the theory of infinite series, (1908 and 1926), Macmillan London, Originally published by
[37] Buchholz, H., The confluent hypergeometric function, (1969), Springer-Verlag Berlin · Zbl 0169.08501
[38] Burkhardt, H., Über den gebrauch divergenter reihen in der zeit von 1750-1860, Math. Ann., 70, 169-206, (1911) · JFM 42.0264.01
[39] Caliceti, E.; Meyer-Hermann, M.; Ribeca, P.; Surzhykov, A.; Jentschura, U. D., From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions, Phys. Rep., 446, 1-96, (2007)
[40] Carleman, T.; Gammel, J. L., Quasi-analytic functions, (1975), Los Alamos Scientific Laboratory Los Alamos, Translation LA-4702-TR
[41] Carlson, B. C., Special functions of applied mathematics, (1977), Academic Press New York · Zbl 0394.33001
[42] Čížek, J.; Vinette, F.; Weniger, E. J., Examples on the use of symbolic computation in physics and chemistry: applications of the inner projection technique and of a new summation method for divergent series, Int. J. Quantum Chem., Symp., 25, 209-223, (1991)
[43] Čížek, J.; Vinette, F.; Weniger, E. J., On the use of the symbolic language Maple in physics and chemistry: several examples, (de Groot, R. A.; Nadrchal, J., Proceedings of the Fourth International Conference on Computational Physics PHYSICS COMPUTING ’92, (1993), World Scientific Singapore), Int. J. Mod. Phys. C, 4, 257-270, (1993), Reprinted in
[44] Čížek, J.; Zamastil, J.; Skála, L., New summation technique for rapidly divergent perturbation series. hydrogen atom in magnetic field, J. Math. Phys., 44, 962-968, (2003) · Zbl 1061.81028
[45] Corless, R. M., Essential Maple 7: an introduction for scientific programmers, (2002), Springer-Verlag New York · Zbl 0987.68563
[46] Costin, O., Asymptotics and Borel summability, (2009), Chapman & Hall/CRC Boca Raton · Zbl 1169.34001
[47] Cuyt, A.; Brevik Petersen, V.; Verdonk, B.; Waadeland, H.; Jones, W. B., Handbook of continued fractions for special functions, (2008), Springer-Verlag New York · Zbl 1150.30003
[48] Cuyt, A.; Wuytack, L., Nonlinear methods in numerical analysis, (1987), North-Holland Amsterdam · Zbl 0609.65001
[49] Cvetič, G.; Yu, J.-Y., Borel-Padé vs Borel-weniger method: a QED and a QCD example, Mod. Phys. Lett. A, 15, 1227-1235, (2000)
[50] Dai, L.; Li, J. X.; Zang, W. P.; Tian, J. G., Vacuum electron acceleration driven by a tightly focused radially polarized Gaussian beam, Opt. Express, 19, 9303-9308, (2011)
[51] Digernes, T.; Varadarajan, V. S., Notes on Euler’s work on divergent factorial series and their associated continued fractions, Indian J. Pure Appl. Math., 41, 39-66, (2010) · Zbl 1203.11023
[52] Driver, K.; Jordaan, K.; Martínez-Finkelshtein, A., Pólya frequency sequences and real zeros of some \({}_3F_2\) polynomials, J. Math. Anal. Appl., 332, 1045-1055, (2007) · Zbl 1114.33006
[53] Drummond, J. E., A formula for accelerating the convergence of a general series, Bull. Aust. Math. Soc., 6, 69-74, (1972) · Zbl 0221.65006
[54] Dutka, J., On the summation of some divergent series of Euler and the zeta functions, Arch. Hist. Exact Sci., 50, 187-200, (1996) · Zbl 0858.01018
[55] Dyson, D. J., Divergence of perturbation theory in quantum electrodynamics, Phys. Rev., 85, 32-33, (1952)
[56] Elliott, D., Truncation errors in Padé approximations to certain functions: an alternative approach, Math. Comput., 21, 398-406, (1967) · Zbl 0158.06701
[57] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher transcendental functions, vol. 2, (1953), McGraw-Hill New York · Zbl 0052.29502
[58] Ferraro, G., The first modern definition of the sum of a divergent series: an aspect of the rise of 20th century mathematics, Arch. Hist. Exact Sci., 54, 101-135, (1999) · Zbl 0932.01026
[59] Ferraro, G., The rise and development of the theory of series up to the early 1820s, (2008), Springer-Verlag New York · Zbl 1141.01007
[60] Ferreira, C.; López, J. L.; Mainar, E.; Temme, N. M., Asymptotic approximations of integrals: an introduction, with recent developments and applications to orthogonal polynomials, Electron. Trans. Numer. Anal., 19, 58-83, (2005) · Zbl 1122.41018
[61] Fields, J. L., Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III, J. Math. Anal. Appl., 12, 593-601, (1965) · Zbl 0136.05502
[62] Fields, J. L.; Luke, Y. L., Asymptotic expansions of a class of hypergeometric polynomials with respect to the order, J. Math. Anal. Appl., 6, 394-403, (1963) · Zbl 0113.28005
[63] Fields, J. L.; Luke, Y. L., Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II, J. Math. Anal. Appl., 7, 440-451, (1963) · Zbl 0126.08503
[64] Fischer, J., On the role of power expansions in quantum field theory, Int. J. Mod. Phys. A, 12, 3625-3663, (1997) · Zbl 0899.22018
[65] Flajolet, P.; Sedgewick, R., Mellin transforms and asymptotics: finite differences and Rice’s integral, Theor. Comput. Sci., 144, 101-124, (1995) · Zbl 0869.68056
[66] Flajolet, P.; Sedgewick, R., Analytic combinatorics, (2009), Cambridge U. P. Cambridge · Zbl 1165.05001
[67] Frobenius, G., Ueber relationen zwischen den Näherungsbrüchen von potenzreihen, J. Reine Angew. Math., 90, 1-17, (1881) · JFM 12.0331.02
[68] Gil, A.; Segura, J.; Temme, N. M., Numerical methods for special functions, (2007), SIAM Philadelphia · Zbl 1144.65016
[69] Gil, A.; Segura, J.; Temme, N. M., Basic methods for computing special functions, (Simos, T. E., Recent Advances in Computational and Applied Mathematics, (2011), Springer-Verlag Dordrecht), 67-121 · Zbl 1216.65034
[70] Graffi, S.; Grecchi, V., Borel summability and indeterminacy of the Stieltjes moment problem: application to the anharmonic oscillators, J. Math. Phys., 19, 1002-1006, (1978) · Zbl 0432.40007
[71] Graves-Morris, P. R.; Roberts, D. E.; Salam, A., The epsilon algorithm and related topics, (Brezinski, C., Numerical Analysis 2000, Vol. 2: Interpolation and Extrapolation, (2000), Elsevier Amsterdam), 122, 51-80, (2000), Reprinted · Zbl 0970.65004
[72] Grecchi, V.; Maioli, M.; Martinez, A., Padé summability of the cubic oscillator, J. Phys. A, 42, 425208, (2009), (17 pp.) · Zbl 1179.81073
[73] Grecchi, V.; Martinez, A., The spectrum of the cubic oscillator, Commun. Math. Phys., 319, 479-500, (2013) · Zbl 1268.81073
[74] Hardy, G. H., Divergent series, (1949), Clarendon Press Oxford · Zbl 0032.05801
[75] Heck, A., Introduction to Maple, (2003), Springer-Verlag New York · Zbl 1020.65001
[76] Ismail, M. E.H., Classical and quantum orthogonal polynomials in one variable, (2005), Cambridge U. P. Cambridge · Zbl 1082.42016
[77] Jentschura, U. D.; Becher, J.; Weniger, E. J.; Soff, G., Resummation of QED perturbation series by sequence transformations and the prediction of perturbative coefficients, Phys. Rev. Lett., 85, 2446-2449, (2000)
[78] Jentschura, U. D.; Gies, H.; Valluri, S. R.; Lamm, D. R.; Weniger, E. J., QED effective action revisited, Can. J. Phys., 80, 267-284, (2002)
[79] Jentschura, U. D.; Lötstedt, E., Numerical calculation of Bessel, Hankel and Airy functions, Comput. Phys. Commun., 183, 506-519, (2012) · Zbl 1264.65028
[80] Jentschura, U. D.; Mohr, P. J.; Soff, G.; Weniger, E. J., Convergence acceleration via combined nonlinear-condensation transformations, Comput. Phys. Commun., 116, 28-54, (1999) · Zbl 0995.81526
[81] Jentschura, U. D.; Weniger, E. J.; Soff, G., Asymptotic improvement of resummations and perturbative predictions in quantum field theory, J. Phys. G, 26, 1545-1568, (2000)
[82] Karlsson, J.; von Sydow, B., The convergence of Padé approximants to series of Stieltjes, Ark. Mat., 14, 43-53, (1976) · Zbl 0323.30045
[83] Kline, M., Mathematical thought from ancient to modern times, (1972), Oxford U. P. Oxford · Zbl 0277.01001
[84] Knopp, K., Theorie und anwendung der unendlichen reihen, (1964), Springer-Verlag Berlin · JFM 57.0249.08
[85] Kozlov, V. V., Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of leonhard Euler), Russ. Math. Surv., 62, 639-661, (2007) · Zbl 1140.01017
[86] (Le Guillou, J. C.; Zinn-Justin, J., Large-Order Behaviour of Perturbation Theory, (1990), North-Holland Amsterdam)
[87] Levin, D., Development of non-linear transformations for improving convergence of sequences, Int. J. Comput. Math. B, 3, 371-388, (1973) · Zbl 0274.65004
[88] Li, J.; Zang, W.; Tian, J., Simulation of Gaussian laser beams and electron dynamics by weniger transformation method, Opt. Express, 17, 4959-4969, (2009)
[89] Li, J.-X.; Zang, W.; Li, Y.-D.; Tian, J., Acceleration of electrons by a tightly focused intense laser beam, Opt. Express, 17, 11850-11859, (2009)
[90] Luke, Y. L., The special functions and their approximations I, (1969), Academic Press New York · Zbl 0193.01701
[91] Luke, Y. L., The special functions and their approximations II, (1969), Academic Press New York · Zbl 0193.01701
[92] Luke, Y. L., Mathematical functions and their approximations, (1975), Academic Press New York · Zbl 0351.33007
[93] Luke, Y. L., On the error in Padé approximations for functions defined by Stieltjes integrals, Comput. Math. Appl., 3, 307-314, (1977) · Zbl 0373.65013
[94] Milne-Thomson, L. M., The calculus of finite differences, (1933), Macmillan London, Originally published by · Zbl 0008.01801
[95] Nielsen, N., Lehrbuch der unendlichen reihen, (1909), Teubner Leipzig and Berlin · JFM 40.0301.02
[96] Nielsen, N., Die gammafunktion, (1906), Teubner Leipzig and Berlin, Originally published by · JFM 36.0500.01
[97] Nörlund, N.-E., Leçons sur LES Séries d’interpolation, (1926), Gautier-Villars Paris · JFM 52.0301.04
[98] Nörlund, N.-E., Leçons sur LES équations linéaires aux différences finies, (1929), Gautier-Villars Paris · JFM 55.0869.01
[99] Nörlund, N. E., Vorlesungen über differenzenrechnung, (1924), Springer-Verlag Berlin, Originally published by · JFM 50.0315.02
[100] (Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions, (2010), Cambridge U. P. Cambridge), Available online under · Zbl 1198.00002
[101] Padé, H., Sur la représentation approachée d’une fonction par des fractions rationelles, Ann. Sci. Éc. Norm. Super., 9, 3-93, (1892) · JFM 24.0360.02
[102] Paris, R. B.; Kaminski, D., Asymptotics and Mellin-Barnes integrals, (2001), Cambridge U. P. Cambridge · Zbl 0983.41019
[103] Poincaré, H., Sur LES intégrales irrégulières des équations linéaires, Acta Math., 8, 295-344, (1886) · JFM 18.0273.02
[104] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical recipes: the art of scientific computing, (2007), Cambridge U. P. Cambridge · Zbl 1132.65001
[105] Richards, D. S.P., Totally positive kernels, polýa frequency functions, and generalized hypergeometric series, Linear Algebra Appl., 137/138, 467-478, (1990) · Zbl 0707.33001
[106] Schoenberg, I. J., On Pólya frequency functions. I. the totally positive functions and their Laplace transforms, J. d’Anal. Math., 1, 331-374, (1951) · Zbl 0045.37602
[107] Shanks, D., Non-linear transformations of divergent and slowly convergent sequences, J. Math. and Phys. (Cambridge, Mass.), 34, 1-42, (1955) · Zbl 0067.28602
[108] Shawyer, B.; Watson, B., Borel’s method of summability, (1994), Oxford U. P. Oxford · Zbl 0840.40001
[109] Sidi, A., A new method for deriving Padé approximants for some hypergeometric functions, J. Comput. Appl. Math., 7, 37-40, (1981) · Zbl 0452.41010
[110] Sidi, A., Practical extrapolation methods, (2003), Cambridge U. P. Cambridge · Zbl 1041.65001
[111] Simon, B., Coupling constant analyticity for the anharmonic oscillator, Ann. Phys. (NY), 58, 76-136, (1970)
[112] Simon, B., Large orders and summability of eigenvalue perturbation theory: a mathematical overview, Int. J. Quant. Chem., 21, 3-25, (1982)
[113] Smith, D. A.; Ford, W. F., Acceleration of linear and logarithmic convergence, SIAM J. Numer. Anal., 16, 223-240, (1979) · Zbl 0407.65002
[114] Smith, D. A.; Ford, W. F., Numerical comparisons of nonlinear convergence accelerators, Math. Comput., 38, 481-499, (1982) · Zbl 0487.65004
[115] Stahl, H., Spurious poles in Padé approximation, J. Comput. Appl. Math., 99, 511-527, (1998) · Zbl 0928.41011
[116] Sternin, B. Y.; Shatalov, V. E., Borel-Laplace transform and asymptotic theory, (1996), CRC Press Boca Raton
[117] Stieltjes, T. J., Recherches sur quelques séries semi-convergentes, Ann. Sci. Éc. Norm. Super., 3, 201-258, (1886) · JFM 18.0197.01
[118] J. Stirling, Methodus differentialis sive tractatus de summatione et interpolatione serierum infinitarum, London, 1730.
[119] Suslov, I. M., Divergent perturbation series, J. Exp. Theor. Phys., 100, 1188-1234, (2005)
[120] Szegö, G., Orthogonal polynomials, (1975), American Mathematical Society Providence, Rhode Island · JFM 65.0278.03
[121] Temme, N. M., Numerical aspects of special functions, Acta Numer., 16, 379-478, (2007) · Zbl 1135.65011
[122] Trefethen, L. N., Approximation theory and approximation practice, (2013), SIAM Philadelphia · Zbl 1264.41001
[123] Tucciarone, J., The development of the theory of summable divergent series from 1880 to 1925, Arch. Hist. Exact Sci., 10, 1-40, (1973) · Zbl 0275.01010
[124] Tweddle, I., James Stirling’s methodus differentialis: an annotated translation of Stirling’s text, (2003), Springer-Verlag London · Zbl 1031.01008
[125] Varadarajan, V. S., Euler and his work on infinite series, Bull. Am. Math. Soc., 44, 515-539, (2007) · Zbl 1135.01010
[126] Weniger, E. J., Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep., 10, 189-371, (1989)
[127] Weniger, E. J., On the summation of some divergent hypergeometric series and related perturbation expansions, J. Comput. Appl. Math., 32, 291-300, (1990) · Zbl 0739.33005
[128] Weniger, E. J., Interpolation between sequence transformations, Numer. Algorithms, 3, 477-486, (1992) · Zbl 0794.65004
[129] Weniger, E. J., On the efficiency of linear but nonregular sequence transformations, (Cuyt, A., Nonlinear Numerical Methods and Rational Approximation II, (1994), Kluwer Dordrecht), 269-282 · Zbl 0807.65001
[130] Weniger, E. J., Verallgemeinerte summationsprozesse als numerische hilfsmittel für quantenmechanische und quantenchemische rechnungen, (1994), Fachbereich Chemie und Pharmazie, Universität Regensburg, Los Alamos preprint
[131] Weniger, E. J., A convergent renormalized strong coupling perturbation expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator, Ann. Phys. (NY), 246, 133-165, (1996) · Zbl 0877.47041
[132] Weniger, E. J., Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations, Comput. Phys., 10, 496-503, (1996)
[133] Weniger, E. J., Construction of the strong coupling expansion for the ground state energy of the quartic, sextic and octic anharmonic oscillator via a renormalized strong coupling expansion, Phys. Rev. Lett., 77, 2859-2862, (1996)
[134] Weniger, E. J., Nonlinear sequence transformations: a computational tool for quantum mechanical and quantum chemical calculations, Int. J. Quant. Chem., 57, 265-280, (1996)
[135] Weniger, E. J., Erratum: nonlinear sequence transformations: a computational tool for quantum mechanical and quantum chemical calculations, Int. J. Quant. Chem., 58, 319-321, (1996)
[136] Weniger, E. J., Performance of superconvergent perturbation theory, Phys. Rev. A, 56, 5165-5168, (1997)
[137] Weniger, E. J., Prediction properties of aitken’s iterated \(\operatorname{\Delta}^2\) process, of wynn’s epsilon algorithm, and of Brezinski’s iterated theta algorithm, (Brezinski, C., Numerical Analysis 2000, Vol. 2: Interpolation and Extrapolation, (2000), Elsevier Amsterdam), 122, 329-356, (2000), Reprinted · Zbl 0974.65002
[138] Weniger, E. J., Irregular input data in convergence acceleration and summation processes: general considerations and some special Gaussian hypergeometric series as model problems, Comput. Phys. Commun., 133, 202-228, (2001) · Zbl 0976.65006
[139] Weniger, E. J., Mathematical properties of a new Levin-type sequence transformation introduced by čížek, zamastil, and skála. I. algebraic theory, J. Math. Phys., 45, 1209-1246, (2004) · Zbl 1070.81056
[140] Weniger, E. J., Asymptotic approximations to truncation errors of series representations for special functions, (Iske, A.; Levesley, J., Algorithms for Approximation, (2007), Springer-Verlag Berlin), 331-348 · Zbl 1117.65042
[141] Weniger, E. J., Further discussion of sequence transformation methods, (2007), Subtopic “Related Resources” (R1) on the Numerical Recipes (third edition) Webnotes page
[142] Weniger, E. J., On the analyticity of Laguerre series, J. Phys. A, 41, 425207, (2008), (43 pp.) · Zbl 1154.30003
[143] Weniger, E. J., An introduction to the topics presented at the conference “approximation and extrapolation of convergent and divergent sequences and series” CIRM luminy: September 28, 2009-October 2, 2009, Appl. Numer. Math., 60, 1184-1187, (2010)
[144] Weniger, E. J., Summation of divergent power series by means of factorial series, Appl. Numer. Math., 60, 1429-1441, (2010) · Zbl 1209.40001
[145] Weniger, E. J., On the mathematical nature of Guseinov’s rearranged one-range addition theorems for Slater-type functions, J. Math. Chem., 50, 17-81, (2012) · Zbl 1317.81121
[146] Weniger, E. J.; Čížek, J., Rational approximations for the modified Bessel function of the second kind, Comput. Phys. Commun., 59, 471-493, (1990) · Zbl 0875.65036
[147] Weniger, E. J.; Čížek, J.; Vinette, F., Very accurate summation for the infinite coupling limit of the perturbation series expansions of anharmonic oscillators, Phys. Lett. A, 156, 169-174, (1991)
[148] Weniger, E. J.; Čížek, J.; Vinette, F., The summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic, sextic, and octic anharmonic oscillators using nonlinear sequence transformations, J. Math. Phys., 34, 571-609, (1993) · Zbl 0794.34045
[149] Weniger, E. J.; Steinborn, E. O., Nonlinear sequence transformations for the efficient evaluation of auxiliary functions for GTO molecular integrals, (Defranceschi, M.; Delhalle, J., Numerical Determination of the Electronic Structure of Atoms, Diatomic and Polyatomic Molecules, NATO ASI Series, (1989), Kluwer Dordrecht), 341-346, Proceedings of the NATO Advanced Research Workshop, Versailles, France, 17-22 April 1988
[150] Widder, D. V., The Stieltjes transform, Trans. Am. Math. Soc., 43, 7-60, (1938) · Zbl 0018.13102
[151] Widder, D. V., The Laplace transform, (1946), Princeton U. P. Princeton · Zbl 0060.24801
[152] Wimp, J., Sequence transformations and their applications, (1981), Academic Press New York · Zbl 0566.47018
[153] Wong, R., Asymptotic approximations of integrals, (1989), Academic Press San Diego · Zbl 0679.41001
[154] Wynn, P., On a device for computing the \(e_m(S_n)\) transformation, Math. Tables Other Aids Comput., 10, 91-96, (1956) · Zbl 0074.04601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.