CALCOLO

, Volume 8, Issue 3, pp 255–272

A transformation of series

Article

DOI: 10.1007/BF02575517

Cite this article as:
Wynn, P. Calcolo (1971) 8: 255. doi:10.1007/BF02575517

Abstract

The transformation
$$\sum\limits_{\nu = 0}^\infty {t_\nu z^\nu \to } \sum\limits_{\nu = 0}^\infty {\left\{ {\sum\limits_{\tau = 0}^{h - 1} {z^\tau } \Delta ^\nu t_{h\nu + \tau } + \frac{{z^h }}{{1 - z}}\Delta ^\nu t_{h(\nu + 1)} } \right\}} \left( {\frac{{z^{h + 1} }}{{1 - z}}} \right)^\nu$$
(*)
whereh≥0 is an integer and Δ operates upon the coefficients {tv} of the series being transformed, is derived. Whenh=0, the above transformation is the generalised Euler transformation, of which (*) is itself a generalisation. Based upon the assumption that\(t_\nu = \int\limits_0^1 {\varrho ^\nu d\sigma (\varrho ) } (\nu = 0, 1,...)\), where σ(ϱ) is bounded and non-decreasing for 0≤ϱ≤1 and subject to further restrictions, a convergence theory of (*) is given. Furthermore, the question as to when (*) functions as a convergence acceleration transformation is investigated. Also the optimal valne ofh to be taken is derived. A simple algorithm for constructing the partial sums of (*) is devised. Numerical illustrations relating to the case in whichtv=(v+1)−1(v=0,1,...) are given.

Copyright information

© IAC 1972

Authors and Affiliations

  • P. Wynn
    • 1
  1. 1.Department of MathematicsLouisiana State University in New OrleansNew Orleans

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