### Abstract

The convergence of large-order expansions in δ= 1/D, where D is the dimensionality of coordinate space, for energies E(δ) of Coulomb systems is strongly affected by singularities at δ= 1 and δ= 0. Padé-Borel approximants with modifications that completely remove the singularities at δ= 1 and remove the dominant singularity at δ= 0 are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function F(δ) = ∑_{j}E′_{j}/j!, where E′_{j} are the the expansion coefficients of the energy with singularity structure removed at δ= 1, farther from the origin and thereby accelerate summation convergence. The ground-state energies of He and H^{+}
_{2} are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using F_{m}(δ) = ∑_{j}E′_{j}/Γ(j + 1 - m) is considered. The standard deviation of results calculated with different values of the shift parameter m is proposed as a measure of summation accuracy.

Original language | English |
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Pages (from-to) | 5112-5122 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 39 |

Issue number | 10 |

Publication status | Published - Oct 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Mathematical Physics*,

*39*(10), 5112-5122.