Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by Č\'ıžek, Zamastil, and Skála. I. Algebraic Theory

Abstract

Č\'ıžek, Zamastil, and Skála [J. Math. Phys. \textbf{44}, 962 - 968 (2003)] introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence $\{s_n \}_{n=0}^{\infty}$ of partial sums, but also explicit estimates $\{ω_n \}_{n=0}^{\infty}$ for the truncation errors. The explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance Padé approximants. Special cases of the new transformation are sequence transformations introduced by Levin [Int. J. Comput. Math. B \textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189 - 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A \textbf{226}, 299 - 349 (1927)]. The algebraic theory of these transformations - explicit expressions, recurrence formulas, explicit expressions in the case of special remainder estimates, and asymptotic order estimates satisfied by rational approximants to power series - is formulated in terms of hitherto unknown mathematical properties of the new transformation introduced by Č\'ıžek, Zamastil, and Skála. This leads to a considerable formal simplification and unification.