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An Analogue for Strong Summability of Abel's Summability Method

Published online by Cambridge University Press:  20 January 2009

C. F. Harington  and
J. M. Hyslop
C. F. Harington
Affiliation:
University of the Witwatersrand, Johannesburg, South Africa.
J. M. Hyslop
Affiliation:
University of the Witwatersrand, Johannesburg, South Africa.
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Given a series Σan, we define , by the relation

where is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,

we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 9 , Issue 1 , October 1953 , pp. 28 - 34
Copyright
Copyright © Edinburgh Mathematical Society 1953

References

REFERENCES

Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge, 1934).Google Scholar
Hyslop, J. M., Proc. Glasgow Math. Assoc., 1 (1951), 16–20.CrossRefGoogle Scholar
Kogbetliantz, E., Bull, des Sciences Math. (2), 49 (1925), 234–256.Google Scholar
Kuttner, B., Journal London Math. Soc., 21 (1946), 118–122.Google Scholar
Marcinkiewicz, J., Journal London Math. Soc., 14 (1939), 162–168.CrossRefGoogle Scholar
Paley, R. E. A. C., Proc. Cambridge Phil. Soc., 26 (1930), 429–437.CrossRefGoogle Scholar
Winn, C. E., Math. Zeitschrift, 37 (1933), 481–492.CrossRefGoogle Scholar