| Item type: | Article | ||||
|---|---|---|---|---|---|
| Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||
| Publisher: | IOP PUBLISHING LTD | ||||
| Place of Publication: | BRISTOL | ||||
| Volume: | 41 | ||||
| Number of Issue or Book Chapter: | 42 | ||||
| Page Range: | p. 425207 | ||||
| Date: | 2008 | ||||
| Institutions: | Chemistry and Pharmacy > Institut für Physikalische und Theoretische Chemie | ||||
| Identification Number: |
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| Keywords: | SLATER-TYPE ORBITALS; NONCENTRAL INTERACTION POTENTIALS; COMPLETE ORTHONORMAL SETS; RANGE ADDITION THEOREMS; UNIFIED ANALYTICAL TREATMENT; FIELD GRADIENT INTEGRALS; GROUND-STATE ENERGY; MULTICENTER MULTIELECTRON INTEGRALS; NONLINEAR SEQUENCE TRANSFORMATIONS; NONSCREENED COULOMB POTENTIALS | ||||
| Dewey Decimal Classification: | 500 Science > 540 Chemistry & allied sciences | ||||
| Status: | Published | ||||
| Refereed: | Yes, this version has been refereed | ||||
| Created at the University of Regensburg: | Yes | ||||
| Item ID: | 67772 |
Abstract
The transformation of a Laguerre series f (z) = Sigma(infinity)(n=0) lambda L-(alpha)(n)n((alpha))(z) to a power series f (z) = Sigma(infinity)(n=0) gamma(n)Z(n) is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power ...
Abstract
The transformation of a Laguerre series f (z) = Sigma(infinity)(n=0) lambda L-(alpha)(n)n((alpha))(z) to a power series f (z) = Sigma(infinity)(n=0) gamma(n)Z(n) is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patterns of the Laguerre series coefficients lambda((alpha))(n) n -> infinity can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients lambda((alpha))(n) either decay exponentially or factorially as n -> infinity. The situation is much more complicated - but also much more interesting - if the lambda((alpha))(n) decay only algebraically as n -> infinity. If the lambda((alpha))(n) ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the lambda((alpha))(n) ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients lambda((alpha))(n) possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulae for hypergeometric series F-p+1(p), but if the lambda((alpha))(n) have a complicated structure or if only their numerical values are available, numerical summation techniques have to be employed. It is shown that certain nonlinear sequence transformations - in particular the so- called delta transformation (Weniger 1989 Comput. Phys. Rep. 10 189 - 371 (equation (8.44)))-are able to sum the divergent series occurring in this context effectively. As a physical application of the results of this paper, the legitimacy of the rearrangement of certain one-range addition theorems for Slater-type functions (Guseinov 1980 Phy. Rev. A 22 369-71, Guseinov 2001 Int. J. Quantum Chem. 81 126-29, Guseinov 2002 Int. J. Quantum Chem. 90 114-8) is investigated.
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