# A transformation of series

Article

- Received:

DOI: 10.1007/BF02575517

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

## Abstract

The transformation where

$$\sum\limits_{\nu = 0}^\infty {t_\nu z^\nu \to } \sum\limits_{\nu = 0}^\infty {\left\{ {\sum\limits_{\tau = 0}^{h - 1} {z^\tau } \Delta ^\nu t_{h\nu + \tau } + \frac{{z^h }}{{1 - z}}\Delta ^\nu t_{h(\nu + 1)} } \right\}} \left( {\frac{{z^{h + 1} }}{{1 - z}}} \right)^\nu$$

(*)

*h*≥0 is an integer and Δ operates upon the coefficients {*t*_{v}} of the series being transformed, is derived. When*h*=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 of*h*to be taken is derived. A simple algorithm for constructing the partial sums of (*) is devised. Numerical illustrations relating to the case in which*t*_{v}*=(v+1)*^{−1}*(v=0,1,...)*are given.## Copyright information

© IAC 1972