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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

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