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On γ-Matrices and their Application to the Binomial Series

Published online by Cambridge University Press:  20 January 2009

P. Vermes
P. Vermes
Affiliation:
Birkbeck College, University of London.
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The known methods of “summing” divergent series, e.g. the means of Cesàro, Riesz, Borel, Lindelöf, Mittag-Leffler are particular cases of the transformation of a sequence (formed from the partial sums) by a T-matrix. An equivalent method is that of the transformation of the series by a γ-matrix, the fundamental properties of which have been proved by Carmichael, Perron and Bosanquet.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 8 , Issue 1 , October 1947 , pp. 1 - 13
Copyright
Copyright © Edinburgh Mathematical Society 1947

References

page 1 note 1 Dienes, P., The Taylor Series (Oxford, 1931), 396–397. This book will be referred to as T.S.Google Scholar

page 1 note 2 T.S. 389.Google Scholar

page 1 note 3 T.S. 399.Google Scholar

page 1 note 4 Vermes, P., “Product of a T-matrix and a γ-matrix.” Journal London Math. Soc. 21 (1946), 129–134 (129).CrossRefGoogle Scholar

page 3 note 1 This remark is due to Mr H. Kestelman.

page 3 note 2 T.S. 393. 396–397.Google Scholar

page 3 note 3 T.S. 411–412.Google Scholar

page 3 note 4 T.S. 418.Google Scholar

page 3 note 5 T.S. 420.Google Scholar

page 4 note 1 See for example [3.IV] of this paper.

page 5 note 1 T.S. 401.Google Scholar

page 5 note 2 T.S. 419–420.Google Scholar

page 6 note 1 T.S. 418.Google Scholar

page 8 note 1 G (1) denotes the first diminutive of G, defined in (3.1).Google Scholar

page 9 note 1 T.S. 399. Theorem VI was used to construct the γ-matrix.

page 11 note 1 Cooke, R. G. and Dienes, P., “On the effective range of generalized limit processes,” Proc. London Math. Soc. (2) 45 (1939), 45–63 (53-55).CrossRefGoogle Scholar