On the summation of some divergent hypergeometric series and related perturbation expansions

Abstract

Divergent hypergeometric series 2 F 0(α,β;−1/ζ) occur frequently in Poincaré-type asymptotic expansions of special functions. These divergent series 2 F 0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied. There is considerable evidence that Levin's sequence transformation (1973) and in particular also some closely related sequence transformations (Weniger (1989)), which were derived recently, sum divergent series 2 F 0 much more efficiently than Padé approximants. Similar summation problems occur also in the case of divergent Rayleigh-Schrödinger perturbation expansions of elementary quantum mechanical systems. A comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that Levin's sequence transformation and the recently derived new sequence transformations are again more efficient than Padé approximants. However, the superiority of Levin's sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion.