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Published online by Cambridge University Press: 18 May 2009
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.
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.
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