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Transactions of the American Mathematical Society

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Convergence acceleration for generalized continued fractions

Authors: Paul Levrie and Lisa Jacobsen
Journal: Trans. Amer. Math. Soc. 305 (1988), 263-275
MSC: Primary 65B05; Secondary 65Q05
MathSciNet review: 920158
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Abstract: The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an $ n$-fraction for which the underlying recurrence relation is of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.

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  • [1] M. G. de Bruin, Convergence of generalized 𝐶-fractions, J. Approx. Theory 24 (1978), no. 3, 177–207. MR 516674, 10.1016/0021-9045(78)90023-0
  • [2] M. G. de Bruin and L. Jacobsen, The dominance concept for linear recurrence relations with applications to continued fractions, Nieuw Arch. Wisk. (4) 3 (1985), no. 3, 253–266. MR 834113
  • [3] -, Modification of generalised continued fractions. I, Lecture Notes in Math., vol 1237 (J. Gilewicz, M. Pindor, W. Siemaszko, Eds.), Springer-Verlag, Berlin, 1987, pp. 161-176.
  • [4] J. R. Cash, A note on the numerical solution of linear recurrence relations, Numer. Math. 34 (1980), no. 4, 371–386. MR 577404, 10.1007/BF01403675
  • [5] P. Van der Cruyssen, Linear difference equations and generalized continued fractions, Computing 22 (1979), no. 3, 269–278 (English, with German summary). MR 620219, 10.1007/BF02243567
  • [6] Lisa Jacobsen, Modified approximants for continued fractions, construction and applications, Norske Vid. Selsk. Skr., no. 3 (1983).
  • [7] P. Kreuser, Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen, Thesis (Tubingen), Borna-Leipzig, 1914.
  • [8] Oskar Perron, Über Summengleichungen und Poincarésche Differenzengleichungen, Math. Ann. 84 (1921), no. 1-2, 1–15 (German). MR 1512016, 10.1007/BF01458689
  • [9] Oskar Perron, Über lineare Differenzengleichungen und eine Anwendung auf lineare Differentialgleichungen mit Polynomkoeffizienten, Math. Z 72 (1959/1960), 16–24 (German). MR 0110902
  • [10] Wolfgang J. Thron and Haakon Waadeland, Accelerating convergence of limit-periodic continued fractions $ K({a_n}/1)$, Numer. Math. 34 (1980), 155-170.
  • [11] W. J. Thron and Haakon Waadeland, Analytic continuation of functions defined by means of continued fractions, Math. Scand. 47 (1980), no. 1, 72–90. MR 600079
  • [12] W. J. Thron and Haakon Waadeland, Convergence questions for limit periodic continued fractions, Rocky Mountain J. Math. 11 (1981), no. 4, 641–657. MR 639449, 10.1216/RMJ-1981-11-4-641
  • [13] William B. Jones, W. J. Thron, and Haakon Waadeland (eds.), Analytic theory of continued fractions, Lecture Notes in Mathematics, vol. 932, Springer-Verlag, Berlin-New York, 1982. Notas de Matemática [Mathematical Notes], 20. MR 690450

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