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1994 | 29 | 1 | 7-18
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Summation of slowly convergent series

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Among the applications of orthogonal polynomials described briefly on my previous visit to this Center [9, §3.2] were slowly convergent series whose terms could be represented in terms of the Laplace transform at integer arguments. We proposed to sum such series by means of Gaussian quadrature rules applied to suitable integrals involving weight functions of Einstein and Fermi type (cf. [13]). In the meantime it transpired that the technique is applicable to a large class of numerical series and, suitably adapted, also to power series and Fourier series of interest in plate problems. In the following we give a summary of these new applications and the contexts in which they arise.
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  • Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907, U.S.A.
  • [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, NBS Appl. Math. Ser. 55, U.S. Government Printing Office, Washington, D.C., 1964.
  • [2] J. Boersma and J. P. Dempsey, On the numerical evaluation of Legendre's chi-function, Math. Comp. 59 (1992), 157-163.
  • [3] W. J. Cody, Chebyshev approximations for the Fresnel integrals, Math. Comp. 22 (1968), 450-453. Loose microfiche suppl. A1-B4.
  • [4] W. J. Cody, K. A. Paciorek and H. C. Thacher, Jr., Chebyshev approximations for Dawson's integral, ibid. 24 (1970), 171-178.
  • [5] P. J. Davis, Spirals: from Theodorus to Chaos, AK Peters, Boston 1993.
  • [6] K. M. Dempsey, D. Liu and J. P. Dempsey, Plana's summation formula for $∑_{m=1,3,...}^∞ m^{-2} sin(mα)$, $m^{-3}cos(mα)$, $m^{-2} A^m$, $m^{-3} A^m$, Math. Comp. 55 (1990), 693-703.
  • [7] B. Gabutti, personal communication, June 1991.
  • [8] W. Gautschi, Minimal solutions of three-term recurrence relations and orthogonal polynomials, Math. Comp. 36 (1981), 547-554.
  • [9] W. Gautschi, Some applications and numerical methods for orthogonal polynomials, in: Numerical Analysis and Mathematical Modelling, A. Wakulicz (ed.), Banach Center Publ. 24, PWN-Polish Scientific Publishers, Warszawa 1990, 7-19.
  • [10] W. Gautschi, Computational aspects of orthogonal polynomials, in: Orthogonal Polynomials - Theory and Practice, P. Nevai (ed.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 294, Kluwer, Dordrecht 1990, 181-216.
  • [11] W. Gautschi, A class of slowly convergent series and their summation by Gaussian quadrature, Math. Comp. 57 (1991), 309-324.
  • [12] W. Gautschi, On certain slowly convergent series occurring in plate contact problems, ibid. 57 (1991), 325-338.
  • [13] W. Gautschi and G. V. Milovanović, Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series, ibid. 44 (1985), 177-190.
  • [14] E. Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math. 89 (1980), 19-44.
  • [15] J. Wimp, Sequence Transformations and Their Application, Math. Sci. Engrg. 154, Academic Press, New York 1981.
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