Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series

Convergence problems occur abundantly in all branches of mathematics or in the mathematical treatment of the sciences. Sequence transformations are principal tools to overcome convergence problems of the kind. They accomplish this by converting a slowly converging or diverging input sequence $\{s_n \}_{n=0}^{\infty}$ into another sequence $\{s^{\prime}_n \}_{n=0}^{\infty}$ with hopefully better numerical properties. Pad\'{e} approximants, which convert the partial sums of a power series to a doubly indexed sequence of rational functions, are the best known sequence transformations, but the emphasis of the review will be on alternative sequence transformations which for some problems provide better results than Pad\'{e} approximants.Comment: 29 pages, LaTeX, 0 figure

Summation of Divergent Power Series by Means of Factorial Series

Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are useful numerical tools for the summation of divergent (inverse) power series. This is documented by summing the divergent asymptotic expansion for the exponential integral $E_{1} (z)$ and the factorially divergent Rayleigh-Schr\"{o}dinger perturbation expansion for the quartic anharmonic oscillator. Stirling numbers play a key role since they occur as coefficients in expansions of an inverse power in terms of inverse Pochhammer symbols and vice versa. It is shown that the relationships involving Stirling numbers are special cases of more general orthogonal and triangular transformations.Comment: Accepted for publication in Applied Numerical Mathematics, 15 pages, LaTeX2e, 0 figure

Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.Comment: 20 pages, LaTeX2e, 0 figure

Comment on "Fourier transform of hydrogen-type atomic orbitals'', Can. J. Phys. Vol.\ 96, 724 - 726 (2018) by N. Y\"{u}k\c{c}\"{u} and S. A. Y\"{u}k\c{c}\"{u}

Podolsky and Pauling (Phys. Rev. \textbf{34}, 109 - 116 (1929)) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Y\"{u}k\c{c}\"{u} and Y\"{u}k\c{c}\"{u}, 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 the Podolsky and Pauling 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.Comment: 15 pages, LaTeX2e, 0 figures Numerical Algorithms, in press (2019

The Spherical Tensor Gradient Operator

The spherical tensor gradient operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$, which is obtained by replacing the Cartesian components of $\bm{r}$ by the Cartesian components of $\nabla$ in the regular solid harmonic ${\mathcal{Y}}_{\ell}^{m} (\bm{r})$, is an irreducible spherical tensor of rank $\ell$. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank $\ell$. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. London Math. Soc. {\bf 24}, 54 - 67 (1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ is applied to them. Fourier transformation is very helpful to understand the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ since it produces ${\mathcal{Y}}_{\ell}^{m} (-\mathrm{i} \bm{p})$. It is also possible to apply ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ to generalized functions, yielding for instance the spherical delta function $\delta_{\ell}^{m} (\bm{r})$. The differential operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of $r^{\nu} {\mathcal{Y}_{\ell}^{m}} (\bm{r})$ with $\nu \in \mathbb{R}$.Comment: 55 pages, LaTeX2e, 0 figure

Addition Theorems as Three-Dimensional Taylor Expansions. II. BB Functions and Other Exponentially Decaying Functions

Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form $\exp (x' \partial /\partial x) \exp (y' \partial /\partial y) \exp (z' \partial /\partial z)$ would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator $\exp (\mathbf{r}' \cdot \mathbf{\nabla})$ involving powers of the Laplacian $\mathbf{\nabla}^2$ and spherical tensor gradient operators $\mathcal{Y}_{\ell}^{m} (\nabla)$, which are irreducible spherical tensors of ranks zero and $\ell$, respectively [F.D.\ Santos, Nucl. Phys. A {\bf 212}, 341 (1973)]. In this way, it is indeed possible to derive addition theorems by doing three-dimensional Taylor expansions [E.J. Weniger, Int. J. Quantum Chem. {\bf 76}, 280 (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of $B$ functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater-type functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of $B$ functions, the addition theorems for these functions can be written down immediately.Comment: 17 pages, LaTeX2e, 0 figures. Submitted to the Per-Olof L\"owdin Honorary Volume, International Journal of Quantum Chemistr

Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives" by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 213)

Addition theorems are principal tools that express a function $f (\bm{r} \pm \bm{r}')$ in terms of products of other functions that only depend on either $\bm{r}$ or $\bm{r}'$. The best known example of such an addition theorem is the Laplace expansion of the Coulomb potential which possesses a characteristic two-range form. Guseinov [Chem. Phys. {\bf 309}, 209 - 213 (2005)] derived one-range addition theorems for the Coulomb potential via the limit $\beta \to 0$ in previously derived one-range addition theorems for the Yukawa potential $\exp \bigl(-\beta | \bm{r}-\bm{r}'| \bigr) /| \bm{r}-\bm{r}'|$. At first sight, this looks like a remarkable achievement, but from a mathematical point of view, Guseinov's work is at best questionable and in some cases fundamentally flawed. One-range addition theorems are expansions in terms of functions that are complete and orthonormal in a given Hilbert space, but Guseinov replaced the complete and orthonormal functions by nonorthogonal Slater-type functions and rearranged the resulting expansions. This is a dangerous operation whose validity must be checked. It is shown that the one-center limit $\bm{r}' = \bm{0}$ of Guseinov's rearranged Yukawa addition theorems as well as of several other addition theorems does not exist. Moreover, the Coulomb potential does not belong to any of the Hilbert spaces implicitly used by Guseinov. Accordingly, one-range addition theorems for the Coulomb potential diverge in the mean. Instead, these one-range addition theorems have to interpreted as expansions of generalized functions in the sense of Schwartz that converge weakly in suitable functionals.Comment: 42 pages, LaTeX2e, 0 figures; references added, minor changes in the text, typos correcte

A Rational Approximant for the Digamma Function

Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Pad\'{e} approximants or other rational functions constructed from sequence transformations. However, neither Pad\'{e} approximants nor sequence transformation utilize the information which is avaliable in the case of a special function -- all power series coefficients as well as the truncation errors are explicitly known -- in an optimal way. Thus, alternative rational approximants, which can profit from additional information of that kind, would be desirable. It is shown that in this way a rational approximant for the digamma function can be constructed which possesses a transformation error given by an explicitly known series expansion.Comment: 11 pages, LaTeX2e, 0 figures. o Appear in the Proceedings (Numerical Algorithms) of the International Conference on Numerical Algorithms, Marrakesh, Morocco, October 1-5, 200

Convergence Analysis of the Summation of the Euler Series by Pad\'e Approximants and the Delta Transformation

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 physics. In this article, we analyze the summation of the Euler series by Pad\'e approximants and the delta transformation [E. J. Weniger, Comput. Phys. Rep. Vol.10, 189 (1989), Eq. (8.4-4)] 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 new factorial series representation of the truncation error of the Euler series [R. Borghi, Appl. Num. Math. Vol.60, 1242 (2010)]. We derive explicit expressions for the transformation errors of Pad\'e approximants and of the delta transformation. A subsequent asymptotic analysis proves \emph{rigorously} the convergence of both Pad\'e and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pad\'e. This is in agreement with previous numerical results.Comment: 46 pages, LaTeX2e, 3 figure

Extrapolation Methods for Improving the Convergence of Oligomer Calculations to the Infinite Chain Limit of Quasi-Onedimensional Stereoregular Polymers

Quasi-onedimensional stereoregular polymers as for example polyacetylene are currently of considerable interest. There are basically two different approaches for doing electronic structure calculations: One method is essentially based on concepts of solid state theory. The other method is essentially a quantum chemical method since it approximates the polymer by oligomers consisting of a finite number of monomer units. In this way, the highly developed technology of quantum chemical molecular programs can be used. Unfortunately, oligomers of finite size are not necessarily able to model those features of a polymer which crucially depend of its in principle infinite extension. In such a case extrapolation techniques can be extremely helpful. For example, one can perform electronic structure calculations for a sequence of oligomers with an increasing number of monomer units. In the next step, one then can try to determine the limit of this sequence for an oligomer of infinite length with the help of suitable extrapolation methods. Several different extrapolation methods are discussed which are able to accomplish an extrapolation of energies and properties of oligomers to the infinite chain limit. Calculations for the ground state energy of polyacetylene are presented which demonstrate the practical usefulness of extrapolation methods.Comment: 29 pages, tables, LaTeX, corrected list of author