Convergence analysis of the summation of the factorially divergent Euler series by Padé approximants and the delta transformation

Borghi, Riccardo; Weniger, Ernst Joachim

doi:10.1016/j.apnum.2015.03.007

Cited by 14 publications

(13 citation statements)

References 120 publications

(186 reference statements)

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“…We also tried to apply some series acceleration techniques that are explained for example in refs. [45][46][47]. 11 Unfortunately, to the best of our knowledge, no transformation that works for our specific problem is known.…”

Section: Jhep03(2016)189mentioning

confidence: 99%

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

Frellesvig

Tommasini

Wever

2016

J. High Energ. Phys.

102

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Abstract:We give expressions for all generalized polylogarithms up to weight four in terms of the functions log, Li n , and Li 2,2 , valid for arbitrary complex variables. Furthermore we provide algorithms for manipulation and numerical evaluation of Li n and Li 2,2 , and add codes in Mathematica and C++ implementing the results. With these results we calculate a number of previously unknown integrals, which we add in appendix C.

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Section: Jhep03(2016)189mentioning

confidence: 99%

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

Frellesvig

Tommasini

Wever

2016

J. High Energ. Phys.

102

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“…The rational function on the right-hand side of (22) is of the same type as the left-hand side of this modified generating function. If we make in (24) the substitutions x ‫ۋ‬ ͑p 2 Ϫ ␤ 2 ͒/͑p 2 ϩ ␤ 2 ͒ and ‫ۋ‬ ℓ ϩ 1, we obtain the following expansion in terms of Gegenbauer polynomials:…”

Section: The Work Of Podolsky and Paulingmentioning

confidence: 99%

Comment on “Fourier transform of hydrogen-type atomic orbitals”

Weniger

2019

Can. J. Phys.

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Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi:10.1103/PhysRev.34.109) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü (Can. ), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling's formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov's functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions. Résumé: Podolsky et Pauling (Phys. Rev. 34, 109 (1929) doi:10.1103/PhysRev.34.109) ont été les premiers à développer une expression explicite pour la transformée de Fourier de la fonction propre d'un état lié de l'hydrogène. Yükçü et Yükçü (Can. ) ont procédé différemment, apparemment sans connaître le travail de Podolsky et Pauling, ni les nombreuses références antérieures sur la transformée de Fourier. Ils ont exprimé un polynôme généralisé de Laguerre comme une somme finie de puissances ou, de façon équivalente, ils ont exprimé la fonction propre d'un état lié de l'hydrogène comme une somme finie de fonctions de type Slater. Cette approche semble très simple, mais elle mène à des expressions comparativement compliquées qui ne peuvent pas égaler la simplicité du résultat classique de Podolsky et Pauling. Il reste néanmoins possible de reproduire, non seulement la formule de Podolsky et Pauling, mais d'obtenir des résultats de qualité similaires pour les transformées de Fourier d'autres fonctions étroitement reliées, comme les Sturmiens, les fonctions Lambda ou de Guseinov, en faisant l'expansion des polynômes généralisés de Laguerre en termes de fonctions appelées fonctions réduites (reduced) de Bessel. [Traduit par la Rédaction]Mots-clés : fonctions propres d'états liés de l'hydrogène, transformation de Fourier, polynômes généralisés de Laguerre, série hypergéométrique, fonctions de type Slater, fonctions réduites de Bessel.

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“…Thus, the corresponding Euler series 2 F 0 (1, 1; −z) is its associated Stieltjes series (see for example [2,Chapter 5]). This Stieltjes property guarantees the convergence of certain subsequences of the Padé table of 2 F 0 (1, 1; −z) to the Euler integral (see for example [2,Chapter 5.2] or the discussion in [13]). It has been shown in many calculations that sequence transformations are principal tools that can accomplish an efficient summation of the factorially divergent expansions of the type of the Euler series (see for example [13] and references therein).…”

Section: Divergent Seriesmentioning

confidence: 99%

“…This Stieltjes property guarantees the convergence of certain subsequences of the Padé table of 2 F 0 (1, 1; −z) to the Euler integral (see for example [2,Chapter 5.2] or the discussion in [13]). It has been shown in many calculations that sequence transformations are principal tools that can accomplish an efficient summation of the factorially divergent expansions of the type of the Euler series (see for example [13] and references therein). However, in the case of most sequence transformations, no rigorous theoretical convergence proofs are known (this applies also to Padé approximants if the series to be transformed is not Stieltjes).…”

Section: Divergent Seriesmentioning

confidence: 99%

“…However, in the case of most sequence transformations, no rigorous theoretical convergence proofs are known (this applies also to Padé approximants if the series to be transformed is not Stieltjes). An exception is the delta transformation [88,] of the Euler series, whose convergence to the Euler integral for all z ∈ C \ [0, ∞) was proved rigorously in [13]. In the following Tables 4.6 and 4.7, we use as input data the partial sums…”

Section: Divergent Seriesmentioning

confidence: 99%

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Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

Chang

et al. 2019

Numer Algor

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We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

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Convergence analysis of the summation of the factorially divergent Euler series by Padé approximants and the delta transformation

Cited by 14 publications

References 120 publications

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

Comment on “Fourier transform of hydrogen-type atomic orbitals”

Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

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