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Note on the Strong Summability of Series

Published online by Cambridge University Press:  18 May 2009

J. M. Hyslop
J. M. Hyslop
Affiliation:
Univeesity of the WitwatebsbandJohannesburg
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Given the series ,the n-th Casáro sum of order k is defined by the relation

where is the binomial coefficient . Let Then Σan is said to be summable (C; K) to the sum s if, as n → ∞, The series is said to be absolutely summable (C; k), or summable | is convergent. The series is said to be strongly summable (C; k) with index p, or summable [Ck, p], to the sum s if

It is assumed that k and p are positive.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 1 , Issue 1 , January 1952 , pp. 16 - 20
Copyright
Copyright © Glasgow Mathematical Journal Trust 1952

References

page 16 note * Kuttner, B., Journal London Math. Soc., 21 (1946), 118–122Google Scholar.

page 17 note * Winn, C. E., Math. Zeitschrift, 37 (1933), 481–492CrossRefGoogle Scholar.

page 17 note † it has been shown that, given any T-matrix, there is a series summable [C; 1, p], p < l, but not summable (T). See B. Kuttner, loc.cit.

Page 17 note ‡ B. Kuttner, loc.cit.

Page 18 note * See Hardy, Littlewood and Pólya, , Inequalities (Cambridge University Press, 1934), 229Google Scholar.

Page 19 note * C. E. Winn, loc.cit.